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Skylanders Academy
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Skylanders Academy
Genre
Action-adventure, comedy, fantasy
Based on
Skylanders
by Activision
Developed by
Eric Rogers
Directed by
Arthur Qwak
Solène Azernour
Voices of
Justin Long
Ashley Tisdale
Jonathan Banks
Chris Diamantopoulos
Jonny Rees
Bobcat Goldthwait
Richard Steven Horvitz
Norm Macdonald
Harland Williams
Felicia Day
Opening theme
"Harmony" by Timbaland Feat. Dalton Diehl
Composer(s)
Conrad Wedde
Samuel Scott
Lukasz Buda
Country of origin
United States
Original language(s)
English
No. of seasons
2 (Renewed for season 3)
No. of episodes
25 (list of episodes)
Production
Executive producer(s)
Bobby Kotick
Eric Rogers
Sander Schwartz
Stacey Sher
Nick van Dyk
Producer(s)
Coco Francini
Emily Levitan
Running time
44 minutes (episode 1)
24-25 minutes
Production company(s)
TeamTO
Activision Blizzard Studios
Release
Original network
Netflix
Original release
October 28, 2016 – present
Skylanders Academy is an American animated web television series produced by TeamTO and Activision Blizzard Studios based on the Skylanders series.[1] The first season debuted on Netflix on October 28, 2016, while a second season debuted on October 6, 2017.[2] A trailer for the series debuted on October 12, 2016.[3]
Contents
[hide]
1 Plot
2 Voice cast
2.1 Main cast
2.2 Recurring voice actors
2.3 Notable guest voices
3 Episodes
3.1 Season 1 (2016)
3.2 Season 2 (2017)
3.3 Season 3 (2018)
4 Production
5 References
6 External links
Plot[edit]
In the world of Skylands, Spyro the Dragon, Stealth Elf, and Eruptor are new graduates at Skylanders Academy. Under the teachings of Master Eon and Jet-Vac, the three of them will learn what it means to be a Skylander while fighting the evil Kaos and other villains of Skylands.
Voice cast[edit]
Main cast[edit]
The voice cast consists of:
Justin Long as Spyro the Dragon[4]
Jason Ritter as Dark Spyro
Ashley Tisdale as Stealth Elf[4]
Jonathan Banks as Eruptor[4]
Chris Diamantopoulos as Master Eon
Jonny Rees as Jet-Vac
Bobcat Goldthwait as Pop Fizz
Richard Steven Horvitz as Kaos
Norm Macdonald as Glumshanks[4]
Harland Williams as Hugo
Felicia Day as Cynder[4] (Season 2)
Recurring voice actors[edit]
John DiMaggio as Chef Pepper Jack
James Hetfield as Wolfgang
Fred Tatasciore as Snap Shot, Strykore
Chris Diamantopoulos as Chompy Mage
John Mathot as Kindly Mabu Prisoner
Catherine O'Hara as Kaossandra/Kaos's Mom
Parker Posey as the Dreamcatcher
Susan Sarandon as Golden Queen
Jim Cummings as Malefor
Jill Talley as Roller Brawl
Courtenay Taylor as Hex
Josh Robert Thompson as Skull
Daniel Wu as King Pen
Billy West as Food Fight, Kaboom
Cedric Yarbrough as Broccoli Guy
Grey Griffin as Sprocket[5]
Notable guest voices[edit]
Eric Rogers as Crash Bandicoot
DanTDM as Cy (Water Imaginator)
John DiMaggio as Bad Breath
Billy West as Bully
Episodes[edit]
Season 1 (2016)[edit]
No. overall
No. in season
Title[6]
Directed by
Written by
Release date[6]
1
1 "Skylanders Unite!" Arthur Qwak Eric Rogers October 28, 2016 (2016-10-28)
In Skylanders Academy, the training institution for new generations of Skylanders, fellow students Spyro the dragon, Stealth Elf, and Eruptor prepare for graduation. Unfortunately, Spyro's oversized ego tempered by Master Eon's constant praise gets in the way of his becoming a Skylander when he throws a party for the whole school and oversleeps, failing his final exam while his friends pass on. Meanwhile, Kaos, longtime enemy of the Skylanders, sneaks into the Academy to locate the map to the Core of Light and steals the Book of Skylanders, freezing all the Skylanders and leaving Spyro the only one able to stop him. Though he is nearly defeated, he manages to unfreeze the book and release the others who manage to defeat Kaos. For his courage and newfound humility, Master Eon makes Spyro a full-fledged Skylander.
2
2 "My Way or the Sky Way" Arthur Qwak Eric Rogers October 28, 2016 (2016-10-28)
The newly formed core Skylander team faces issues when Spyro's impulsive attitude clashes with Jet-Vac's rigid "Skylander Way" approach. Jet-Vac decides to move into their house to try and drill Spyro into a more refined Skylander, but his method of thorough planning causes more problems when they arrive almost too late to save the Falling Forest from burning. Eon reminds Jet-Vac that instinct is just as important as training for a Skylander, and Jet-Vac and Spyro reconcile their differences in time to stop Kaos, attempting to use a hypnotized Hugo to infiltrate the Relics Room disguised as a restaurant mascot, and the team celebrates by making Jet-Vac's move-in permanent.
3
3 "Missing Links" Arthur Qwak
Co-Director: Solène Azernour Josh Haber October 28, 2016 (2016-10-28)
The Skylanders have formed into a team, but their teamwork is shown to be poor as they consistently fail to pass a basic Fire Viper training test. To help them get to the heart of this issue, Eon's old friend Sensei King Pen shows them how to overcome their obstacle by giving them the opportunity to build a new teammate, the result being a banana-headed Skylander named Cy. Though he is equipped with all of their powers, Cy still struggles as each of the Skylanders show him how to use their powers insisting their style is superior. This leads to more of the same when a real Fire Viper attacks the Academy, leading the Skylanders to realize they must combine their powers together and finally attain victory.
4
4 "Dream Girls" Arthur Qwak
Co-Director: Solène Azernour Josh Haber, Joanna Lewis, and Kristine Songco October 28, 2016 (2016-10-28)
Stealth Elf is selected to give a SED (Skills, Elements & Defense) Talk to the cadets at the Academy, though she expresses doubt that she is capable of giving worthy advice. Meanwhile, everyone at the Academy, including her teammates, suddenly share in Hugo's irrational fear of sheep, leading Stealth Elf to devote herself to discovering the cause. She soon finds it in Dreamcatcher, who embraces evil and uses her powers to plague dreams, but Stealth Elf uses her skill and cunning to defeat her at her own game in the Dream Realm. Dreamcatcher is expelled from the Academy and arrested along with Kaos, once again trying to sneak in, while Elf returns in time to give a rousing speech to the Academy.
5
5 "The Hole Truth" Arthur Qwak
Co-Director: Solène Azernour Dean Stefan October 28, 2016 (2016-10-28)
As per regulations, Spyro, Stealth Elf, and Eruptor are assigned as guards at Cloudcracker Prison, home to the worst criminals in the Skylands, including the infamous Doom Raiders led by the Golden Queen. The Doom Raiders decide to use the tense trio to their advantage and trick them into sending each of them into 'The Hole,' the maximum-security area in the prison. Though the Skylanders manage to realize the deception, the Doom Raiders are still able to combine their evil powers and escape the prison with Kaos tagging along. The three are regretful but resolve to recapture the Doom Raiders before they can wreak havoc on the Skylands.
6
6 "Space Invaders" Arthur Qwak
Co-Director: Solène Azernour Duane Capizzi October 28, 2016 (2016-10-28)
The Doom Raiders have taken refuge in Kaos's castle and berate him more by the second when they take over. Meanwhile, the Skylanders and Eon go on a training retreat in the Falling Forest to improve their combat skills and teamwork, but it slowly turns into a camping trip to Stealth Elf's dismay. They are soon interrupted by an astral projection of Kaos's head that captures Stealth Elf, trapping her inside Kaos's mind. Stealth Elf controls Kaos into sneaking back into the Academy where the others are grieving over her assumed demise, and manages to convince them to help her escape. Kaos, who had won some slight respect from the Doom Raiders thanks to Stealth Elf's control, retreats and returns to being demeaned.
7
7 "Anger Mismanagement" Arthur Qwak
Co-Director: Solène Azernour Eric Rogers and Josh Haber October 28, 2016 (2016-10-28)
Eruptor's short fuse causes problems with the team's missions when he blunders their latest attempt to capture the Doom Raider Chef Pepper Jack. Eon and the others convince him to go to anger management in order to put his fiery temper under control so that he will be more 'in balance.' Unfortunately, Pepper Jack infiltrates the Skylanders' annual Chili Chomp contest and incapacitates all the Skylanders with a poisoned chili while he goes to blow up the Academy. It is only when Eruptor taps back into his inner fire that he manages to stop Pepper Jack and save the Academy, and he finally gains proper control of his emotions.
8
8 "Pop Rocks" Arthur Qwak
Co-Director: Solène Azernour Jesse Porter October 28, 2016 (2016-10-28)
The Skylanders plan to attend the Skylands Music Festival to stop Wolfgang, another Doom Raider, from using his evil music to transform the crowd into an army of mindless drones. Pop Fizz, who the young Skylanders learn was once in a band with Wolfgang before they chose different paths, is particularly determined to stop him, but the potion he concocts to defend them from Wolfgang's spell instead reverts his memories to before he became a Skylander. Wolfgang uses this to his advantage and tricks Pop into believing they still have their band in order to easily gain access to the stage and enact his plan. Pop however regains his memory and defeats Wolfgang in a music battle, sending the Doom Raider running and leaving the Skylanders to enjoy the festival.
9
9 "Beard Science" Arthur Qwak
Co-Director: Solène Azernour Josh Haber October 28, 2016 (2016-10-28)
Kaos continues to be unsuccessful in his attempts to get the attention of the Golden Queen, who seems to have an infatuation with hair. He uses a spell to gain a new hairdo and ends up stealing Master Eon's beloved beard, the source of his magical power, and instantly gains respect from everyone. At the same time, Eon is rendered completely invisible and forgotten by everyone at the Academy, and with no power or ability to command respect, he decides to leave the Academy. He manages to catch Kaos with his beard, which has caught the eye of the Golden Queen, and manages to take it back in a magical duel. With his beard, the Skylanders notice him and Kaos is rejected by the Queen, returning everything to normal.
10
10 "The Skylands Are Falling!" Arthur Qwak
Co-Director: Solène Azernour Eric Rogers and Brittany Jo Flores October 28, 2016 (2016-10-28)
When the Skylanders fail to stop Bomb Shell from destroying Eon's bronzed beard statue, Spyro convinces Pop Fizz to build the "Evil Scope," a machine that uses a lens enchanted by an unwitting Kaos to detect evil before it can be enacted. Spyro obsessively uses the scope to stop evil all over the Skylands, leaving the other Skylanders and even villains across the Skylands to enjoy more complacent lives, but the wayward evil soon clusters together into a giant black ball that threatens to destroy the Skylands and traps Spyro inside. Realizing that his actions have cause a great imbalance, Spyro escapes the ball, imploding it, but its destruction accidentally brings Crash Bandicoot to their world.
11
11 "Crash Landing" Arthur Qwak
Co-Director: Solène Azernour Eric Rogers and Brittany Jo Flores October 28, 2016 (2016-10-28)
After Crash Bandicoot is somehow transported into their world, the Skylanders have to help him get home by finding a dark relic. While searching for the Relic, Spyro begins to idolize and even attempts to imitate Crash. The Skylanders were able to find the Relic in the form of Kaossandra's evil book, which Kaos stole in order to gain its power, but needed an evil villain to open it; the one he chose being Chompy Mage. The Skylanders are able to get the book and return back to the academy where Eon uses it to summon a portal back to Crash's world, however using it corrupts Eon with darkness.
12
12 "Assault on Skylander Academy" Arthur Qwak
Co-Director: Solène Azernour Eric Rogers October 28, 2016 (2016-10-28)
When Eon falls seriously ill after using the book of darkness, the Skylanders must race to find the Core of Light to use its power to heal him. Meanwhile, the Doom Raiders attack the academy, but are pushed back by the cadets. Kaossandra arrives and strikes a deal with the Golden Queen- she takes back her book and the Doomraiders keep the academy. They succeed in finding the book and Eon, but the Golden Queen goes back on her word and keeps it. Kaos is forced to choose sides and chooses to side with the Golden Queen, forcing Kaossandra to kick him out. She uses a powerful spell to knock out the Doomraiders and leave with Glumshanks, but is forced to leave the book behind. The Doomraiders, with Kaos, retreat as the Skylanders return with the Core of Light's energy to cure Eon. Fully healed, Eon says he'll lock the book in a safe place where nobody can find it, but in truth returns it to Kaossandra as the two are secretly allied in using it to keep a much more powerful and sinister evil locked in a dark realm.
Season 2 (2017)[edit]
No. overall
No. in season
Title[6]
Directed by
Written by
Release date[6]
13
1 "Spyromania" Arthur Qwak and Solène Azernour Ian Weinreich and Clayton Sakoda October 6, 2017 (2017-10-06)
Spyro wants to prove that he has his own superpower like other Skylanders, and he's going to need it when he gets into a dangerous situation.
14
2 "I Dream of Ninjini" Arthur Qwak and Solène Azernour Julia Yorks October 6, 2017 (2017-10-06)
Master Eon tells Stealth Elf to seek training from an elusive ninja named Ninjini, a once-great warrior who doubts her current abilities.
15
3 "Return to Cynder" Arthur Qwak and Solène Azernour Nick Confalone October 6, 2017 (2017-10-06)
A new cadet named Cynder becomes fast friends with Spyro at the Academy, but she questions whether she has what it takes to be a true hero.
16
4 "Thankstaking For the Memories" Arthur Qwak and Solène Azernour Nick Confalone October 6, 2017 (2017-10-06)
Spyro and Stealth Elf get locked in the Grand Library while everyone else is gone for a long holiday weekend.
17
5 "Elementary, My Dear Eruptor" Arthur Qwak and Solène Azernour Jim Martin October 6, 2017 (2017-10-06)
Eruptor discovers he has the Sherlock Holmes-like ability to figure out who's guilty of committing evil deeds.
18
6 "Split Decision" Arthur Qwak and Solène Azernour Natalie Hazen October 6, 2017 (2017-10-06)
When Skull decides to break away from Hex, Hex turns evil and tries to transform the others at the Academy into zombie-like drones.
19
7 "The People vs. Pop Fizz" Arthur Qwak and Solène Azernour Jesse Porter October 6, 2017 (2017-10-06)
Pop Fizz is arrested and put on trial for committing robberies all over the Skylands, and the other Skylanders have to help find the real culprits.
20
8 "One Flu Over the Skylander’s Nest" Arthur Qwak and Solène Azernour Story by: Eric Rogers
Teleplay by: Patrick Rigney and Eric Rogers October 6, 2017 (2017-10-06)
Eon comes to the team house to lead training, but the house is quarantined and everyone ends up getting sick.
21
9 "Belly of the Beast" Arthur Qwak and Solène Azernour Ian Weinreich and Clayton Sakoda October 6, 2017 (2017-10-06)
While fighting the evil Fire Viper, the gang ends up inside its belly and must put aside personal differences to find a way out.
22
10 "Who’s Your Daddy?" Arthur Qwak and Solène Azernour Joanna Lewis and Kristine Songco October 6, 2017 (2017-10-06)
When one of Jet-Vac's eggs hatches, he must try to be a good father to the baby inside. Kaos and Glumshanks believe they might be father and son.
23
11 "Sheep(ball) Dreams" Arthur Qwak and Solène Azernour Brittany Jo Flores October 6, 2017 (2017-10-06)
Dreamcatcher gets amnesia, giving the Skylanders a chance to bring her back to the Academy and prevent her from returning to her evil ways.
24
12 "It Techs Two" Arthur Qwak and Solène Azernour Johanna Richie October 6, 2017 (2017-10-06)
Jet-Vac feels insecure about his role with the Skylanders when Sprocket is brought in and starts taking care of tech-related problems.
25
13 "Touch of Evil" Arthur Qwak and Solène Azernour Eric Rogers October 6, 2017 (2017-10-06)
As he tries to learn more about his ancestors, Spyro searches for a dragon king, Malefor, but Malefor isn't one to talk and beats the dragon to a pulp and uses him as a bargaining chip to get his daughter Cynder back. Kaos attempts to free an evil being named Strykore, whom claims to be the father of Kaos and is seeking to conquer Skylands. Kaos obtains power from Strykcore and uses it to get his power back from his mother and take hers as well, forcing her to runaway on a Hydra. However, the portal will only open when he absorbs the light of a powerful Skylander, with Kaos 'volunteering' Spyro. He strikes a deal with Malefor, who knows who Strykcore is, and gives Spyro over in exchange for being on Strykecores good side and probable power gain. Cynder and Spyro's team arrive to demand Spyro back from Malefor, defeating him easily, but when Malefor plays coy, apologizing to his daughter for all his wrongdoings, he takes her and leaves. Kaos brings Spyro to Strykcore and he absorbs his power, destroying the prison and freeing him, and also turning Spyro evil. The team returns to the Academy just as Kaossandra arrives, revealing what happened and asking for help. Eon decides to reveal to the truth to everyone, taking off his helm to reveal the same mark on his forehead as Kaos, suggesting they may be related.
Season 3 (2018)[edit]
A third season has also been announced, scheduled to begin production in February of 2018.[4]
Production[edit]
At an Investor Day presentation on November 6, 2015, Activision Blizzard announced the formation of Activision Blizzard Studios, a film production subsidiary dedicated to creating original films and television series. Headed by former The Walt Disney Company executive Nick van Dyk, Activision Blizzard Studios would look to produce and adapting Skylanders into a film and television series; the latter being called Skylanders Academy, which started airing on October 28, 2016 on Netflix.[7]
The series is a spin-off, said to be separate from the storyline of the games with no direct tie-ins to sequels. In addition, the different voice cast is made to give the show its own separate identity. However, Richard Horvitz, Jonny Rees, Bobcat Goldthwait, Billy West, Fred Tatasciore, and Courtenay Taylor are the only actors from the games to reprise their roles as Kaos, Jet-Vac, Pop Fizz, Food Fight, Snap Shot, and Hex respectively.[8]
The trailer for the second season was released on September 19, 2017.[9]
References[edit]
Jump up ^ Steinberg, Brian (June 16, 2016). "Netflix Readies Animated 'Spy Kids,' 'Llama Llama' Series (Exclusive)". Variety. Retrieved February 22, 2018.
Jump up ^ Gettell, Oliver (September 20, 2016). "Netflix's Skylanders Academy series sets October premiere date". Entertainment Weekly. Retrieved February 22, 2018.
Jump up ^ Serrao, Nivea (October 12, 2016). "Skylanders Academy trailer introduces Netflix's newest heroes". Entertainment Weekly. Retrieved February 22, 2018.
^ Jump up to: a b c d e f Andreeva, Nellie (February 2, 2017). "'Skylanders Academy' Activision Blizzard Animated Series Renewed For Season 3 By Netflix Ahead Of Season 2 Debut". Deadline. Retrieved February 22, 2018.
Jump up ^ Milligan, Mercedes (September 20, 2017). "Trailer: 'Skylanders Academy' Season 2 Takes Flight Oct. 6". Animation Magazine. Retrieved February 22, 2018.
^ Jump up to: a b c d "Skylanders Academy". Netflix. Retrieved October 28, 2016.
Jump up ^ "Activision Blizzard Launches Activision Blizzard Studios to Create Original Film and TV Content Based on Iconic, Globally-Recognized Franchises". BusinessWire.com. November 6, 2015.
Jump up ^ Eric Rogers [@EricRogersLA] (July 22, 2016). "@pokegirlsky the same reason we chose a lot of different actors for several roles - to give the show its own separate identity" (Tweet) – via Twitter.
Jump up ^ Trumbore, Dave (September 19, 2017). "'Skylanders Academy' Season 2 Trailer: More High-Flying Adventures with New Allies". Collider. Retrieved February 22, 2018.
External links[edit]
Skylanders Academy official website
Skylanders Academy on Netflix
Skylanders Academy on IMDb
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List of OEIS sequences
From Wikipedia, the free encyclopedia
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This article provides a list of integer sequences in the On-Line Encyclopedia of Integer Sequences that have their own English Wikipedia entries.
OEIS link
Name
First elements
Short description
A000002 Kolakoski sequence {1, 2, 2, 1, 1, 2, 1, 2, 2, 1, …} The nth term describes the length of the nth run
A000010 Euler's totient function φ(n) {1, 1, 2, 2, 4, 2, 6, 4, 6, 4, …} φ(n) is the number of positive integers not greater than n that are prime to n.
A000027 Natural numbers {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, …} The natural numbers (positive integers) n ∈ ℕ.
A000032 Lucas numbers L(n) {2, 1, 3, 4, 7, 11, 18, 29, 47, 76, …} L(n) = L(n − 1) + L(n − 2) for n ≥ 2, with L(0) = 2 and L(1) = 1.
A000040 Prime numbers pn {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, …} The prime numbers pn, with n ≥ 1.
A000041 Partition numbers
Pn {1, 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...} The partition numbers, number of additive breakdowns of n.
A000045 Fibonacci numbers F(n) {0, 1, 1, 2, 3, 5, 8, 13, 21, 34, …} F(n) = F(n − 1) + F(n − 2) for n ≥ 2, with F(0) = 0 and F(1) = 1.
A000058 Sylvester's sequence {2, 3, 7, 43, 1807, 3263443, 10650056950807, 113423713055421844361000443, …} a(n + 1) = a(n)⋅a(n − 1)⋅ ⋯ ⋅a(0) + 1 = a(n)2 − a(n) + 1 for n ≥ 1, with a(0) = 2.
A000073 Tribonacci numbers {0, 1, 1, 2, 4, 7, 13, 24, 44, 81, …} T(n) = T(n − 1) + T(n − 2) + T(n − 3) for n ≥ 3, with T(0) = 0 and T(1) = T(2) = 1.
A000108 Catalan numbers Cn {1, 1, 2, 5, 14, 42, 132, 429, 1430, 4862, …} C n = 1 n + 1 ( 2 n n ) = ( 2 n ) ! ( n + 1 ) ! n ! = ∏ k = 2 n n + k k , n ≥ 0. {\displaystyle C_{n}={\frac {1}{n+1}}{2n \choose n}={\frac {(2n)!}{(n+1)!\,n!}}=\prod \limits _{k=2}^{n}{\frac {n+k}{k}},\quad n\geq 0.}
A000110 Bell numbers Bn {1, 1, 2, 5, 15, 52, 203, 877, 4140, 21147, …} Bn is the number of partitions of a set with n elements.
A000111 Euler zigzag numbers En {1, 1, 1, 2, 5, 16, 61, 272, 1385, 7936, …} En is the number of linear extensions of the "zig-zag" poset.
A000124 Lazy caterer's sequence {1, 2, 4, 7, 11, 16, 22, 29, 37, 46, …} The maximal number of pieces formed when slicing a pancake with n cuts.
A000129 Pell numbers Pn {0, 1, 2, 5, 12, 29, 70, 169, 408, 985, …} a(n) = 2a(n − 1) + a(n − 2) for n ≥ 2, with a(0) = 0, a(1) = 1.
A000142 Factorials n! {1, 1, 2, 6, 24, 120, 720, 5040, 40320, 362880, …} n! := 1⋅2⋅3⋅4⋅ ⋯ ⋅n for n ≥ 1, with 0! = 1 (empty product).
A000203 Divisor function σ(n) {1, 3, 4, 7, 6, 12, 8, 15, 13, 18, 12, 28, …} σ(n) := σ1(n) is the sum of divisors of a positive integer n.
A000215 Fermat numbers Fn {3, 5, 17, 257, 65537, 4294967297, 18446744073709551617, 340282366920938463463374607431768211457, …} Fn = 22n + 1 for n ≥ 0.
A000217 Triangular numbers t(n) {0, 1, 3, 6, 10, 15, 21, 28, 36, 45, …} t(n) = C(n + 1, 2) = n (n + 1)/2 = 1 + 2 + ⋯ + n for n ≥ 1, with t(0) = 0 (empty sum).
A000290 Square numbers n2 {0, 1, 4, 9, 16, 25, 36, 49, 64, 81, ...} n2 = n × n
A000292 Tetrahedral numbers T(n) {0, 1, 4, 10, 20, 35, 56, 84, 120, 165, …} T(n) is the sum of the first n triangular numbers, with T(0) = 0 (empty sum).
A000330 Square pyramidal numbers {0, 1, 5, 14, 30, 55, 91, 140, 204, 285, …} n (n + 1)(2n + 1)/6 : The number of stacked spheres in a pyramid with a square base.
A000396 Perfect numbers {6, 28, 496, 8128, 33550336, 8589869056, 137438691328, 2305843008139952128, …} n is equal to the sum s(n) = σ(n) − n of the proper divisors of n.
A000578 Cube numbers n3 {0, 1, 8, 27, 64, 125, 216, 343, 512, 729, ...} n3 = n × n × n
A000668 Mersenne primes {3, 7, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, …} 2p − 1 is prime, where p is a prime.
A000793 Landau's function {1, 1, 2, 3, 4, 6, 6, 12, 15, 20, …} The largest order of permutation of n elements.
A000796 Decimal expansion of π {3, 1, 4, 1, 5, 9, 2, 6, 5, 3, …} Ratio of a circle's circumference to its diameter.
A000930 Narayana's cows {1, 1, 1, 2, 3, 4, 6, 9, 13, 19, …} The number of cows each year if each cow has one cow a year beginning its fourth year.
A000931 Padovan sequence {1, 1, 1, 2, 2, 3, 4, 5, 7, 9, …} P(n) = P(n − 2) + P(n − 3) for n ≥ 3, with P(0) = P(1) = P(2) = 1.
A000945 Euclid–Mullin sequence {2, 3, 7, 43, 13, 53, 5, 6221671, 38709183810571, 139, …} a(1) = 2; a(n + 1) is smallest prime factor of a(1) a(2) ⋯ a(n) + 1.
A000959 Lucky numbers {1, 3, 7, 9, 13, 15, 21, 25, 31, 33, …} A natural number in a set that is filtered by a sieve.
A001006 Motzkin numbers {1, 1, 2, 4, 9, 21, 51, 127, 323, 835, …} The number of ways of drawing any number of nonintersecting chords joining n (labeled) points on a circle.
A001045 Jacobsthal numbers {0, 1, 1, 3, 5, 11, 21, 43, 85, 171, 341, …} a(n) = a(n − 1) + 2a(n − 2) for n ≥ 2, with a(0) = 0, a(1) = 1.
A001065 Sum of proper divisors s(n) {0, 1, 1, 3, 1, 6, 1, 7, 4, 8, …} s(n) = σ(n) − n is the sum of the proper divisors of the positive integer n.
A001113 Decimal expansion of e {2, 7, 1, 8, 2, 8, 1, 8, 2, 8, …} Euler's number in base 10.
A001190 Wedderburn–Etherington numbers {0, 1, 1, 1, 2, 3, 6, 11, 23, 46, …} The number of binary rooted trees (every node has out-degree 0 or 2) with n endpoints (and 2n − 1 nodes in all).
A001263 Narayana numbers {1, 1, 1, 1, 3, 1, 1, 6, 6, 1, 1, 10, 20, 10, 1, …} 1 k ( n − 1 k − 1 ) ( n k − 1 ) {\displaystyle {1 \over k}{n-1 \choose k-1}{n \choose k-1}} read by rows.
A001358 Semiprimes {4, 6, 9, 10, 14, 15, 21, 22, 25, 26, …} Products of two primes, not necessarily distinct.
A001462 Golomb sequence {1, 2, 2, 3, 3, 4, 4, 4, 5, 5, …} a(n) is the number of times n occurs, starting with a(1) = 1.
A001608 Perrin numbers Pn {3, 0, 2, 3, 2, 5, 5, 7, 10, 12, …} P(n) = P(n−2) + P(n−3) for n ≥ 3, with P(0) = 3, P(1) = 0, P(2) = 2.
A001620 Euler–Mascheroni constant γ {5, 7, 7, 2, 1, 5, 6, 6, 4, 9, …} γ = lim n → ∞ ( ∑ k = 1 n 1 k − ln  ( n ) ) = lim b → ∞ ∫ 1 b ( 1 ⌊ x ⌋ − 1 x ) d x . {\displaystyle \gamma =\lim _{n\rightarrow \infty }\left(\sum _{k=1}^{n}{\frac {1}{k}}-\ln(n)\right)=\lim _{b\rightarrow \infty }\int _{1}^{b}\left({1 \over \lfloor x\rfloor }-{1 \over x}\right)\,dx.}
A001622 Decimal expansion of the golden ratio φ {1, 6, 1, 8, 0, 3, 3, 9, 8, 8, …} φ = 1 + √5/2 = 1.6180339887... in base 10.
A002064 Cullen numbers Cn {1, 3, 9, 25, 65, 161, 385, 897, 2049, 4609, 10241, 22529, 49153, 106497, …} Cn = n⋅2n + 1, with n ≥ 0.
A002110 Primorials pn# {1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690, 223092870, …} pn#, the product of the first n primes.
A002113 Palindromic numbers {0, 1, 2, 3, 4, 5, 6, 7, 8, 9, …} A number that remains the same when its digits are reversed.
A002182 Highly composite numbers {1, 2, 4, 6, 12, 24, 36, 48, 60, 120, …} A positive integer with more divisors than any smaller positive integer.
A002193 Decimal expansion of √2 {1, 4, 1, 4, 2, 1, 3, 5, 6, 2, …} Square root of 2.
A002201 Superior highly composite numbers {2, 6, 12, 60, 120, 360, 2520, 5040, 55440, 720720, …} A positive integer n for which there is an e > 0 such that d(n)/ne ≥ d(k)/ke for all k > 1.
A002378 Pronic numbers {0, 2, 6, 12, 20, 30, 42, 56, 72, 90, …} 2t(n) = n (n + 1), with n ≥ 0.
A002559 Markov numbers {1, 2, 5, 13, 29, 34, 89, 169, 194, …} Positive integer solutions of x2 + y2 + z2 = 3xyz.
A002808 Composite numbers {4, 6, 8, 9, 10, 12, 14, 15, 16, 18, …} The numbers n of the form xy for x > 1 and y > 1.
A002858 Ulam number {1, 2, 3, 4, 6, 8, 11, 13, 16, 18, …} a(1) = 1; a(2) = 2; for n > 2, a(n) is least number > a(n − 1) which is a unique sum of two distinct earlier terms; semiperfect.
A002997 Carmichael numbers {561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, 29341, …} Composite numbers n such that an − 1 ≡ 1 (mod n) if a is prime to n.
A003261 Woodall numbers {1, 7, 23, 63, 159, 383, 895, 2047, 4607, …} n⋅2n − 1, with n ≥ 1.
A003459 Permutable primes {2, 3, 5, 7, 11, 13, 17, 31, 37, 71, …} The numbers for which every permutation of digits is a prime.
A005044 Alcuin's sequence {0, 0, 0, 1, 0, 1, 1, 2, 1, 3, 2, 4, 3, 5, 4, 7, 5, 8, 7, 10, 8, 12, 10, 14, …} Number of triangles with integer sides and perimeter n.
A005100 Deficient numbers {1, 2, 3, 4, 5, 7, 8, 9, 10, 11, …} Positive integers n such that σ(n) < 2n.
A005101 Abundant numbers {12, 18, 20, 24, 30, 36, 40, 42, 48, 54, …} Positive integers n such that σ(n) > 2n.
A005114 Untouchable numbers {2, 5, 52, 88, 96, 120, 124, 146, 162, 188, …} Cannot be expressed as the sum of all the proper divisors of any positive integer.
A005150 Look-and-say sequence {1, 11, 21, 1211, 111221, 312211, 13112221, 1113213211, 31131211131221, 13211311123113112211, …} A = 'frequency' followed by 'digit'-indication.
A005224 Aronson's sequence {1, 4, 11, 16, 24, 29, 33, 35, 39, 45, …} "t" is the first, fourth, eleventh, ... letter in this sentence, not counting spaces or commas.
A005235 Fortunate numbers {3, 5, 7, 13, 23, 17, 19, 23, 37, 61, …} The smallest integer m > 1 such that pn# + m is a prime number, where the primorial pn# is the product of the first n prime numbers.
A005349 Harshad numbers in base 10 {1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, …} A Harshad number in base 10 is an integer that is divisible by the sum of its digits (when written in base 10).
A005384 Sophie Germain primes {2, 3, 5, 11, 23, 29, 41, 53, 83, 89, …} A prime number p such that 2p + 1 is also prime.
A005835 Semiperfect numbers {6, 12, 18, 20, 24, 28, 30, 36, 40, 42, …} A natural number n that is equal to the sum of all or some of its proper divisors.
A006037 Weird numbers {70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, …} A natural number that is abundant but not semiperfect.
A006842 Farey sequence numerators {0, 1, 0, 1, 1, 0, 1, 1, 2, 1, …}
A006843 Farey sequence denominators {1, 1, 1, 2, 1, 1, 3, 2, 3, 1, …}
A006862 Euclid numbers {2, 3, 7, 31, 211, 2311, 30031, 510511, 9699691, 223092871, …} pn# + 1, i.e. 1 + product of first n consecutive primes.
A006886 Kaprekar numbers {1, 9, 45, 55, 99, 297, 703, 999, 2223, 2728, …} X2 = Abn + B, where 0 < B < bn and X = A + B.
A007304 Sphenic numbers {30, 42, 66, 70, 78, 102, 105, 110, 114, 130, …} Products of 3 distinct primes.
A007318 Pascal's triangle {1, 1, 1, 1, 2, 1, 1, 3, 3, 1, …} Pascal's triangle read by rows.
A007588 Stella octangula numbers {0, 1, 14, 51, 124, 245, 426, 679, 1016, 1449, 1990, 2651, 3444, 4381, …} Stella octangula numbers: n (2n2 − 1), with n ≥ 0.
A007770 Happy numbers {1, 7, 10, 13, 19, 23, 28, 31, 32, 44, …} The numbers whose trajectory under iteration of sum of squares of digits map includes 1.
A007947 Radical of an integer {1, 2, 3, 2, 5, 6, 7, 2, 3, 10, …} The radical of a positive integer n is the product of the distinct prime numbers dividing n.
A010060 Prouhet–Thue–Morse constant {0, 1, 1, 0, 1, 0, 0, 1, 1, 0, …} τ = ∑ i = 0 ∞ t i 2 i + 1 . {\displaystyle \tau =\sum _{i=0}^{\infty }{\frac {t_{i}}{2^{i+1}}}.}
A014080 Factorions {1, 2, 145, 40585, …} A natural number that equals the sum of the factorials of its decimal digits.
A014577 Regular paperfolding sequence {1, 1, 0, 1, 1, 0, 0, 1, 1, 1, …} At each stage an alternating sequence of 1s and 0s is inserted between the terms of the previous sequence.
A016114 Circular primes {2, 3, 5, 7, 11, 13, 17, 37, 79, 113, …} The numbers which remain prime under cyclic shifts of digits.
A018226 Magic numbers {2, 8, 20, 28, 50, 82, 126, …} A number of nucleons (either protons or neutrons) such that they are arranged into complete shells within the atomic nucleus.
A019279 Superperfect numbers {2, 4, 16, 64, 4096, 65536, 262144, 1073741824, 1152921504606846976, 309485009821345068724781056, …} Positive integers n for which σ2(n) = σ(σ(n)) = 2n.
A027641 Bernoulli numbers Bn {1, -1, 1, 0, -1, 0, 1, 0, -1, 0, 5, 0, -691, 0, 7, 0, -3617, 0, 43867, 0, …}
A031214 First elements in all OEIS sequences {1, 1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, …} One of sequences referring to the OEIS itself.
A033307 Decimal expansion of Champernowne constant {1, 2, 3, 4, 5, 6, 7, 8, 9, 1, …} Formed by concatenating the positive integers.
A034897 Hyperperfect numbers {6, 21, 28, 301, 325, 496, 697, …} k-hyperperfect numbers, i.e. n for which the equality n = 1 + k (σ(n) − n − 1) holds.
A035513 Wythoff array {1, 2, 4, 3, 7, 6, 5, 11, 10, 9, …} A matrix of integers derived from the Fibonacci sequence.
A036262 Gilbreath's conjecture {2, 1, 3, 1, 2, 5, 1, 0, 2, 7, …} Triangle of numbers arising from Gilbreath's conjecture.
A037274 Home prime {1, 2, 3, 211, 5, 23, 7, 3331113965338635107, 311, 773, …} For n ≥ 2, a(n) is the prime that is finally reached when you start with n, concatenate its prime factors (A037276) and repeat until a prime is reached; a(n) = − 1 if no prime is ever reached.
A046075 Undulating numbers {101, 121, 131, 141, 151, 161, 171, 181, 191, 202, …} A number that has the digit form ababab.
A050278 Pandigital numbers {1023456789, 1023456798, 1023456879, 1023456897, 1023456978, 1023456987, 1023457689, 1023457698, 1023457869, 1023457896, …} Numbers containing the digits 0–9 such that each digit appears exactly once.
A052486 Achilles numbers {72, 108, 200, 288, 392, 432, 500, 648, 675, 800, …} Positive integers which are powerful but imperfect.
A060006 Decimal expansion of Pisot–Vijayaraghavan number {1, 3, 2, 4, 7, 1, 7, 9, 5, 7, …} Real root of x3 − x − 1.
A076336 Sierpinski numbers {78557, 271129, 271577, 322523, 327739, 482719, 575041, 603713, 903983, 934909, …} Odd k for which { k⋅2n + 1 : n ∈ ℕ } consists only of composite numbers.
A076337 Riesel numbers {509203, 762701, 777149, 790841, 992077, …} Odd k for which { k⋅2n − 1 : n ∈ ℕ } consists only of composite numbers.
A086747 Baum–Sweet sequence {1, 1, 0, 1, 1, 0, 0, 1, 0, 1, …} a(n) = 1 if the binary representation of n contains no block of consecutive zeros of odd length; otherwise a(n) = 0.
A090822 Gijswijt's sequence {1, 1, 2, 1, 1, 2, 2, 2, 3, 1, …} The nth term counts the maximal number of repeated blocks at the end of the subsequence from 1 to n-1
A094683 Juggler sequence {0, 1, 1, 5, 2, 11, 2, 18, 2, 27, …} If n ≡ 0 (mod 2) then ⌊√n⌋ else ⌊n3/2⌋.
A097942 Highly totient numbers {1, 2, 4, 8, 12, 24, 48, 72, 144, 240, …} Each number k on this list has more solutions to the equation φ(x) = k than any preceding k.
A100264 Decimal expansion of Chaitin's constant {0, 0, 7, 8, 7, 4, 9, 9, 6, 9, …} Chaitin constant (Chaitin omega number) or halting probability.
A104272 Ramanujan primes {2, 11, 17, 29, 41, 47, 59, 67, …} The nth Ramanujan prime is the least integer Rn for which π(x) − π(x/2) ≥ n, for all x ≥ Rn.
A122045 Euler numbers {1, 0, −1, 0, 5, 0, −61, 0, 1385, 0, …} 1 cosh  t = 2 e t + e − t = ∑ n = 0 ∞ E n n ! ⋅ t n . {\displaystyle {\frac {1}{\cosh t}}={\frac {2}{e^{t}+e^{-t}}}=\sum _{n=0}^{\infty }{\frac {E_{n}}{n!}}\cdot t^{n}.}
A1238591 Polite numbers {3, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, …} A positive integer that can be written as the sum of two or more consecutive positive integers.
References[edit]
OEIS core sequences
External links[edit]
Index to OEIS
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1/2 − 1/4 + 1/8 − 1/16 + ⋯
1/2 + 1/4 + 1/8 + 1/16 + ⋯
1/4 + 1/16 + 1/64 + 1/256 + ⋯
1 + 1/1s + 1/2s+ 1/3s + ... (Riemann zeta function)
Divergent
1 + 1 + 1 + 1 + ⋯
1 + 2 + 3 + 4 + ⋯
1 + 2 + 4 + 8 + ⋯
1 − 1 + 1 − 1 + ⋯ (Grandi's series)
Infinite arithmetic series
1 − 2 + 3 − 4 + ⋯
1 − 2 + 4 − 8 + ⋯
1 + 1/2 + 1/3 + 1/4 + ⋯ (harmonic series)
1 − 1 + 2 − 6 + 24 − 120 + ⋯ (alternating factorials)
1/2 + 1/3 + 1/5 + 1/7 + 1/11 + ⋯ (inverses of primes)
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text:
Minoan Bull-leaper
From Wikipedia, the free encyclopedia
Jump to: navigation, search
Bronze Group of a Bull and Acrobat
Material
Bronze
Size
L: 15.5 centimetres (6.1 in)
H: 11.4 centimetres (4.5 in)
W: 4.7 centimetres (1.9 in)
Created
1600BC-1450BC
Period/culture
Late Minoan I
Place
Rethymnon, Crete
Present location
G12/1, British Museum, London
Registration
1966,0328.1
The Minoan bull leaper is a bronze group of a bull and leaper in the British Museum. It is the only known largely complete three-dimensional sculpture depicting Minoan bull-leaping.[1] Although bull leaping certainly took place in Crete at this time, the leap depicted is practically impossible and it has therefore been speculated that the sculpture may be an exaggerated depiction.[1][2] This speculation has been backed up by the testaments of modern-day bull leapers from France and Spain.[3]
Contents
[hide]
1 Description
2 Background
3 Bull-leaping
3.1 Size of ancient Minoan bull leaping bulls
4 Media appearances
5 See also
6 Notes
7 References
8 Further reading
9 External links
Description[edit]
The Minoan bull leaper from the front.
The group was cast in a single mould using the lost-wax casting technique. The group's homogeneity was demonstrated by analysing the composition of the bronze of bull and leaper: both contain about 96% copper and 1.5% tin, with 1% zinc.[4] The small amount of tin in the alloy would have made it difficult for the bronze to fill the mould, resulting in the missing lower legs of the leaper, and probably the arms.[2] Stylistically, the group is coherent, since the arched back of the leaper mirrors the flying gallop posture of the bull.
Background[edit]
Arthur Evans, the excavator of Knossos, first published this object in the Journal of Hellenic Studies.[5] Evans dated the bull-leaper to the Late Minoan I period, so that this object dates to approximately 1600 BC.[6]
It was acquired by the British Museum in 1966 as part of the collection of Captain Edward George Spencer-Churchill (1876-1964). He acquired it in Crete in 1921.[7] The object was widely known before its acquisition and display in the Museum. It was illustrated in several general books[8][9][10] and exhibited at the British Academy in 1936.[11]
Bull-leaping[edit]
Bull leaping and bulls in general are believed to have been an important part of Minoan culture; excavations at Knossos have revealed several frescos depicting bull-leaping.[12] It has been suggested bulls may have had some religious significance to them for example the large, exaggerated size of the bull compared to the human leaper may give an idea of the Minoans' reverence for the power of the animals.[1]
This object has been central to discussions of bull-leaping, since Arthur Evans used it as the basis for his reconstruction of the mechanics of the leap: the leaper grabs the bull's horns, executes a back flip onto the bull's back and then dismounts.[5] As John Younger has pointed out, although this reconstruction has become part of bull-leaping in the popular imagination, comparatively few Minoan depictions show exactly this schema. The majority show the leaper diving over the bull's horns onto the back.[13]
Size of ancient Minoan bull leaping bulls[edit]
Archaeological evidence has now uncovered that the type of bull used by ancient Minoan bull leapers was a cross breed giant aurochs bull, now extinct in Europe. It had a shoulder height of over 6 ft (180 cm) and a hoof size similar to the size of a human head.[14][verification needed]
Media appearances[edit]
The bull-leaper was object 18[15] in the BBC Radio 4 Series A History of the World in 100 Objects. Week 4 of the series was 'The Beginning of Science and Literature': Neil MacGregor emphasised the importance of bronze, used to manufacture tools as well as art objects, and its central role in Mediterranean trade in the Bronze Age.
The BBC promoted A History of the World in 100 Objects with television adverts. One of these was a 30-second commercial featuring a Spanish recortador (bull-leaper) created by the ad agency Fallon Worldwide.[16]
See also[edit]
Bull-Leaping Fresco
Notes[edit]
^ Jump up to: a b c "A History of the World - Object: A bronze statue of an acrobat leaping over a bull from the island of Crete.". BBC and British Museum. 22 September 2008. Retrieved 6 June 2010.
^ Jump up to: a b "Minoan bull leaper". British Museum. Retrieved 5 June 2010.
Jump up ^ "A history of the world in 100 objects". 1. Episode 18. February 10, 2010. BBC. Radio 4.
Jump up ^ Craddock 1976, pp. 93–113
^ Jump up to: a b Evans 1921, pp. 247–259
Jump up ^ "Collections Database Online". British Museum. Retrieved 4 June 2010.
Jump up ^ Higgins, R. (1967). "Recent Acquisitions by the British Museum". Archaeological Reports. 13 (13): 47–52. JSTOR 580975. doi:10.2307/580975.
Jump up ^ Hall 1928, p. 172, fig. 221&222
Jump up ^ Hutchinson 1962, p. 265, pl.16
Jump up ^ Lamb 1929, p. 27, pl.6
Jump up ^ Myres 1936, pp. 9, 17
Jump up ^ "Knossos - Minoan Art work". Retrieved 5 June 2010.
Jump up ^ Younger 1976
Jump up ^ https://www.youtube.com/watch?v=ev-WeyQjCWk
Jump up ^ "A History of the World in 100 Objects". Radio Times. Retrieved 2010-06-06.
Jump up ^ "History of the World". Fallon. Retrieved 6 June 2010.
References[edit]
Craddock, Paul (1976), "The Composition of the Copper Alloys used by the Greek, Etruscan and Roman Civilizations. 1. The Greeks before the Archaic Period", Journal of Archaeological Science, 3 (2): 93–113, doi:10.1016/0305-4403(76)90079-0
Evans, Arthur (1921), "On a Minoan Bronze Group of a Galloping Bull and Acrobatic Figure from Crete. With Glyptic Comparisons and a Note on the Oxford Relief Showing the Taurokathapsia", Journal of Hellenic Studies, 41: 247–259, ISSN 0075-4269, JSTOR 625500, OCLC 486621953, doi:10.2307/625500
Hall, H R (1928), The Civilization of Greece in the Bronze Age (The Rhind Lectures 1923), Methuen, OCLC 493998189
Hutchinson, R W (1962), Prehistoric Crete, Pelican books, A501, Penguin Books, OCLC 1683258
Lamb, Winifred (1929), Greek and Roman bronzes, Methuen's handbooks of archaeology., Methuen, OCLC 1936690
Myres, John (1936), British Archaeological Discoveries in Greece and Crete 1886-1936. Catalogue of the Exhibition Arranged to Commemorate the Fiftieth Anniversary of the British School at Athens, London: Royal Academy of Arts, OCLC 602565099
Younger, John (1976), "Bronze Age Representations of Bull-Leaping", American Journal of Archaeology, 80 (2): 125–137
Further reading[edit]
Evans, Arthur (1921–35), The Palace of Minos at Knossos, Macmillan, OCLC 459432466
Fitton, J Lesley (2002), Minoans, Peoples of the past (London, England), British Museum Press, OCLC 606932988
Higgins, Reynold (1967), Minoan and Mycenaean Art, Thames & Hudson, OCLC 250899895
Higgins, Reynold Alleyne (1970), The Greek Bronze Age, British Museum, ISBN 978-0-7141-1233-6
Hood, Sinclair (1978), The arts in prehistoric Greece, The Pelican history of art, Harmondsworth, ISBN 978-0-14-056142-5
Pendlebury, J D S (1963), The archaeology of Crete, an introduction, New York, Biblo and Tannen, OCLC 609210218
External links[edit]
Wikimedia Commons has media related to Minoan Bull-leaper.
Minoan and Mycenaean at DMOZ
http://projectsx.dartmouth.edu/classics/history/bronze_age/lessons/les/14.html
BBC audio file A History of the World in 100 Objects
McInerney, J., "Bulls and Bull-leaping in the Minoan World", Expedition Magazine 53.3 (December 2011): n. pag.

Preceded by
17: Rhind Mathematical Papyrus A History of the World in 100 Objects
Object 18 Succeeded by
19: Mold cape
This article is about an item held in the British Museum. The object reference is 1966,0328.1.
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text:
Diatomic molecule
From Wikipedia, the free encyclopedia
Jump to navigation Jump to search
A space-filling model of the diatomic molecule dinitrogen, N2
Diatomic molecules are molecules composed of only two atoms, of the same or different chemical elements. The prefix di- is of Greek origin, meaning "two". If a diatomic molecule consists of two atoms of the same element, such as hydrogen (H2) or oxygen (O2), then it is said to be homonuclear. Otherwise, if a diatomic molecule consists of two different atoms, such as carbon monoxide (CO) or nitric oxide (NO), the molecule is said to be heteronuclear.
A periodic table showing the elements that exist as homonuclear diatomic molecules under typical laboratory conditions.
The only chemical elements that form stable homonuclear diatomic molecules at standard temperature and pressure (STP) (or typical laboratory conditions of 1 bar and 25 °C) are the gases hydrogen (H2), nitrogen (N2), oxygen (O2), fluorine (F2), and chlorine (Cl2).[1]
The noble gases (helium, neon, argon, krypton, xenon, and radon) are also gases at STP, but they are monatomic. The homonuclear diatomic gases and noble gases together are called "elemental gases" or "molecular gases", to distinguish them from other gases that are chemical compounds.[2]
At slightly elevated temperatures, the halogens bromine (Br2) and iodine (I2) also form diatomic gases.[3] All halogens have been observed as diatomic molecules, except for astatine, which is uncertain.
The mnemonics BrINClHOF, pronounced "Brinklehof",[citation needed] and HONClBrIF, pronounced "Honkelbrif",[4] and HOFBrINCl (pronouced as Hofbrinkle) have been coined to aid recall of the list of diatomic elements.
Other elements form diatomic molecules when evaporated, but these diatomic species repolymerize when cooled. Heating ("cracking") elemental phosphorus gives diphosphorus, P2. Sulfur vapor is mostly disulfur (S2). Dilithium (Li2) is known in the gas phase. Ditungsten (W2) and dimolybdenum (Mo2) form with sextuple bonds in the gas phase. The bond in a homonuclear diatomic molecule is non-polar. Dirubidium is diatomic.
Contents
1 Heteronuclear molecules
2 Occurrence
3 Molecular geometry
4 Historical significance
5 Excited electronic states
6 Energy levels
6.1 Translational energies
6.2 Rotational energies
6.3 Vibrational energies
6.4 Comparison between rotational and vibrational energy spacings
7 Hund's cases
8 See also
9 References
10 Further reading
11 External links
Heteronuclear molecules[edit]
All other diatomic molecules are chemical compounds of two different elements. Many elements can combine to form heteronuclear diatomic molecules, depending on temperature and pressure.
Common examples include the gases carbon monoxide (CO), nitric oxide (NO), and hydrogen chloride (HCl).
Many 1:1 binary compounds are not normally considered diatomic because they are polymeric at room temperature, but they form diatomic molecules when evaporated, for example gaseous MgO, SiO, and many others.
Occurrence[edit]
Hundreds of diatomic molecules have been identified[5] in the environment of the Earth, in the laboratory, and in interstellar space. About 99% of the Earth's atmosphere is composed of two species of diatomic molecules: nitrogen (78%) and oxygen (21%). The natural abundance of hydrogen (H2) in the Earth's atmosphere is only of the order of parts per million, but H2 is the most abundant diatomic molecule in the universe. The interstellar medium is, indeed, dominated by hydrogen atoms.
Molecular geometry[edit]
Main article: Molecular geometry
Historical significance[edit]
Diatomic elements played an important role in the elucidation of the concepts of element, atom, and molecule in the 19th century, because some of the most common elements, such as hydrogen, oxygen, and nitrogen, occur as diatomic molecules. John Dalton's original atomic hypothesis assumed that all elements were monatomic and that the atoms in compounds would normally have the simplest atomic ratios with respect to one another. For example, Dalton assumed water's formula to be HO, giving the atomic weight of oxygen as eight times that of hydrogen[6], instead of the modern value of about 16. As a consequence, confusion existed regarding atomic weights and molecular formulas for about half a century.
As early as 1805, Gay-Lussac and von Humboldt showed that water is formed of two volumes of hydrogen and one volume of oxygen, and by 1811 Amedeo Avogadro had arrived at the correct interpretation of water's composition, based on what is now called Avogadro's law and the assumption of diatomic elemental molecules. However, these results were mostly ignored until 1860, partly due to the belief that atoms of one element would have no chemical affinity toward atoms of the same element, and also partly due to apparent exceptions to Avogadro's law that were not explained until later in terms of dissociating molecules.
At the 1860 Karlsruhe Congress on atomic weights, Cannizzaro resurrected Avogadro's ideas and used them to produce a consistent table of atomic weights, which mostly agree with modern values. These weights were an important prerequisite for the discovery of the periodic law by Dmitri Mendeleev and Lothar Meyer.[7]
Excited electronic states[edit]
Diatomic molecules are normally in their lowest or ground state, which conventionally is also known as the X {\displaystyle X} state. When a gas of diatomic molecules is bombarded by energetic electrons, some of the molecules may be excited to higher electronic states, as occurs, for example, in the natural aurora; high-altitude nuclear explosions; and rocket-borne electron gun experiments.[8] Such excitation can also occur when the gas absorbs light or other electromagnetic radiation. The excited states are unstable and naturally relax back to the ground state. Over various short time scales after the excitation (typically a fraction of a second, or sometimes longer than a second if the excited state is metastable), transitions occur from higher to lower electronic states and ultimately to the ground state, and in each transition results a photon is emitted. This emission is known as fluorescence. Successively higher electronic states are conventionally named A {\displaystyle A} , B {\displaystyle B} , C {\displaystyle C} , etc. (but this convention is not always followed, and sometimes lower case letters and alphabetically out-of-sequence letters are used, as in the example given below). The excitation energy must be greater than or equal to the energy of the electronic state in order for the excitation to occur.
In quantum theory, an electronic state of a diatomic molecule is represented by
2 S + 1 Λ ( v ) {\displaystyle ^{2S+1}\Lambda (v)}
where S {\displaystyle S} is the total electronic spin quantum number, Λ {\displaystyle \Lambda } is the total electronic angular momentum quantum number along the internuclear axis, and v {\displaystyle v} is the vibrational quantum number. Λ {\displaystyle \Lambda } takes on values 0, 1, 2, …, which are represented by the electronic state symbols Σ {\displaystyle \Sigma } , Π {\displaystyle \Pi } , Δ {\displaystyle \Delta } ,…. For example, the following table lists the common electronic states (without vibrational quantum numbers) along with the energy of the lowest vibrational level ( v = 0 {\displaystyle v=0} ) of diatomic nitrogen (N2), the most abundant gas in the Earth's atmosphere.[9] In the table, the subscripts and superscripts after Λ {\displaystyle \Lambda } give additional quantum mechanical details about the electronic state.
State
Energy ( T 0 {\displaystyle T_{0}} , cm−1)
See note below
X 1 Σ g + {\displaystyle X^{1}\Sigma _{g}^{+}} 0.0
A 3 Σ u + {\displaystyle A^{3}\Sigma _{u}^{+}} 49754.8
B 3 Π g {\displaystyle B^{3}\Pi _{g}} 59306.8
W 3 Δ u {\displaystyle W^{3}\Delta _{u}} 59380.2
B ′ 3 Σ u − {\displaystyle B'^{3}\Sigma _{u}^{-}} 65851.3
a ′ 1 Σ u − {\displaystyle a'^{1}\Sigma _{u}^{-}} 67739.3
a 1 Π g {\displaystyle a^{1}\Pi _{g}} 68951.2
w 1 Δ u {\displaystyle w^{1}\Delta _{u}} 71698.4
Note: The "energy" units in the above table are actually the reciprocal of the wavelength of a photon emitted in a transition to the lowest energy state. The actual energy can be found by multiplying the given statistic by the product of c (the speed of light) and h (Planck's constant), i.e., about 1.99 × 10−25 Joule metres, and then multiplying by a further factor of 100 to convert from cm−1 to m−1.
The aforementioned fluorescence occurs in distinct regions of the electromagnetic spectrum, called "emission bands": each band corresponds to a particular transition from a higher electronic state and vibrational level to a lower electronic state and vibrational level (typically, many vibrational levels are involved in an excited gas of diatomic molecules). For example, N2 A {\displaystyle A} - X {\displaystyle X} emission bands (a.k.a. Vegard-Kaplan bands) are present in the spectral range from 0.14 to 1.45 μm (micrometres).[8] A given band can be spread out over several nanometers in electromagnetic wavelength space, owing to the various transitions that occur in the molecule's rotational quantum number, J {\displaystyle J} . These are classified into distinct sub-band branches, depending on the change in J {\displaystyle J} .[10] The R {\displaystyle R} branch corresponds to Δ J = + 1 {\displaystyle \Delta J=+1} , the P {\displaystyle P} branch to Δ J = − 1 {\displaystyle \Delta J=-1} , and the Q {\displaystyle Q} branch to Δ J = 0 {\displaystyle \Delta J=0} . Bands are spread out even further by the limited spectral resolution of the spectrometer that is used to measure the spectrum. The spectral resolution depends on the instrument's point spread function.
Energy levels[edit]
The molecular term symbol is a shorthand expression of the angular momenta that characterize the electronic quantum states of a diatomic molecule, which are eigenstates of the electronic molecular Hamiltonian. It is also convenient, and common, to represent a diatomic molecule as two point masses connected by a massless spring. The energies involved in the various motions of the molecule can then be broken down into three categories: the translational, rotational, and vibrational energies.
Translational energies[edit]
The translational energy of the molecule is given by the kinetic energy expression:
E t r a n s = 1 2 m v 2 {\displaystyle E_{trans}={\frac {1}{2}}mv^{2}}
where m {\displaystyle m} is the mass of the molecule and v {\displaystyle v} is its velocity.
Rotational energies[edit]
Classically, the kinetic energy of rotation is
E r o t = L 2 2 I {\displaystyle E_{rot}={\frac {L^{2}}{2I}}\,}
where
L {\displaystyle L\,} is the angular momentum
I {\displaystyle I\,} is the moment of inertia of the molecule
For microscopic, atomic-level systems like a molecule, angular momentum can only have specific discrete values given by
L 2 = l ( l + 1 ) ℏ 2 {\displaystyle L^{2}=l(l+1)\hbar ^{2}\,}
where l {\displaystyle l} is a non-negative integer and ℏ {\displaystyle \hbar } is the reduced Planck constant.
Also, for a diatomic molecule the moment of inertia is
I = μ r 0 2 {\displaystyle I=\mu r_{0}^{2}\,}
where
μ {\displaystyle \mu \,} is the reduced mass of the molecule and
r 0 {\displaystyle r_{0}\,} is the average distance between the centers of the two atoms in the molecule.
So, substituting the angular momentum and moment of inertia into Erot, the rotational energy levels of a diatomic molecule are given by:
E r o t = l ( l + 1 ) ℏ 2 2 μ r 0 2 l = 0 , 1 , 2 , . . . {\displaystyle E_{rot}={\frac {l(l+1)\hbar ^{2}}{2\mu r_{0}^{2}}}\ \ \ \ \ l=0,1,2,...\,}
Vibrational energies[edit]
Another type of motion of a diatomic molecule is for each atom to oscillate—or vibrate—along the line connecting the two atoms. The vibrational energy is approximately that of a quantum harmonic oscillator:
E v i b = ( n + 1 2 ) ℏ ω n = 0 , 1 , 2 , . . . . {\displaystyle E_{vib}=\left(n+{\frac {1}{2}}\right)\hbar \omega \ \ \ \ \ n=0,1,2,....\,}
where
n {\displaystyle n} is an integer
ℏ {\displaystyle \hbar } is the reduced Planck constant and
ω {\displaystyle \omega } is the angular frequency of the vibration.
Comparison between rotational and vibrational energy spacings[edit]
The spacing, and the energy of a typical spectroscopic transition, between vibrational energy levels is about 100 times greater than that of a typical transition between rotational energy levels.
Hund's cases[edit]
Main article: Hund's cases
The good quantum numbers for a diatomic molecule, as well as good approximations of rotational energy levels, can be obtained by modeling the molecule using Hund's cases.
See also[edit]
Symmetry of diatomic molecules
AXE method
Octatomic element
Covalent bond
Industrial gas
References[edit]
Jump up ^ Hammond, C.R. (2012). "Section 4: Properties of the Elements and Inorganic Compounds". Handbook of Chemistry and Physics (PDF).
Jump up ^ Emsley, J. (1989). The Elements. Oxford: Clarendon Press. pp. 22–23.
Jump up ^ Whitten, Kenneth W.; Davis, Raymond E.; Peck, M. Larry; Stanley, George G. (2010). Chemistry (9th ed.). Brooks/Cole, Cengage Learning. pp. 337–338.
Jump up ^ Sherman, Alan (1992). Chemistry and Our Changing World. Prentice Hall. p. 82. ISBN 9780131315419.
Jump up ^ Huber, K. P.; Herzberg, G. (1979). Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules. New York: Van Nostrand: Reinhold.
Jump up ^ Langford, Cooper Harold; Beebe, Ralph Alonzo (1995-01-01). The Development of Chemical Principles. Courier Corporation. ISBN 9780486683591.
Jump up ^ Ihde, Aaron J. (1961). "The Karlsruhe Congress: A centennial retrospective". Journal of Chemical Education. 38 (2): 83–86. Bibcode:1961JChEd..38...83I. doi:10.1021/ed038p83. Retrieved 2007-08-24.
^ Jump up to: a b Gilmore, Forrest R.; Laher, Russ R.; Espy, Patrick J. (1992). "Franck-Condon Factors, r-Centroids, Electronic Transition Moments, and Einstein Coefficients for Many Nitrogen and Oxygen Band Systems". Journal of Physical and Chemical Reference Data. 21 (5): 1005–1107. Bibcode:1992JPCRD..21.1005G. doi:10.1063/1.555910.
Jump up ^ Laher, Russ R.; Gilmore, Forrest R. (1991). "Improved Fits for the Vibrational and Rotational Constants of Many States of Nitrogen and Oxygen". Journal of Physical and Chemical Reference Data. 20 (4): 685–712. Bibcode:1991JPCRD..20..685L. doi:10.1063/1.555892.
Jump up ^ Levine, Ira N. (1975), Molecular Spectroscopy, John Wiley & Sons, pp. 508–9, ISBN 0-471-53128-6
Further reading[edit]
Huber, K. P.; Herzberg, G. (1979). Molecular Spectra and Molecular Structure IV. Constants of Diatomic Molecules. New York: Van Nostrand: Reinhold.
Tipler, Paul (1998). Physics For Scientists and Engineers: Vol. 1 (4th ed.). W. H. Freeman. ISBN 1-57259-491-8.
External links[edit]
Hyperphysics – Rotational Spectra of Rigid Rotor Molecules
Hyperphysics – Quantum Harmonic Oscillator
3D Chem – Chemistry, Structures, and 3D Molecules
IUMSC – Indiana University Molecular Structure Center
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Triatomic
Aluminium hydroxide
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Amino radical
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CCP radical
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E-Cyanomethanimine
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Eight
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Nine
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values:
interlanguage-link interwiki-sl
display: none;

values:
interlanguage-link interwiki-tr
display: none;

values: Двоатомна молекула – Ukrainian

values:
interlanguage-link interwiki-uk
display: none;

values: All languages (initial selection from common choices by you and others)

values: /static/images/wikimedia-button-1.5x.png 1.5x, /static/images/wikimedia-button-2x.png 2x

values: /static/images/poweredby_mediawiki_132x47.png 1.5x, /static/images/poweredby_mediawiki_176x62.png 2x

values:
suggestions
display: none; font-size: 13px;

values: mediawiki ltr sitedir-ltr mw-hide-empty-elt ns-0 ns-subject page-Diatomic_molecule rootpage-Diatomic_molecule skin-vector action-view