"Hyperperfect number In mathematics, a ""k""-hyperperfect number is a natural number ""n"" for which the equality ""n"" = 1 + ""k""(""σ""(""n"") − ""n"" − 1) holds, where ""σ""(""n"") is the divisor function (i.e., the sum of all positive divisors of ""n""). A hyperperfect number is a ""k""-hyperperfect number for some integer ""k"". Hyperperfect numbers generalize perfect numbers, which are 1-hyperperfect. The first few numbers in the sequence of ""k""-hyperperfect numbers are 6, 21, 28, 301, 325, 496, 697, ... , with the corresponding values of ""k"" being 1, 2, 1, 6, 3, 1, 12, ... . The first few ""k""-hyperperfect"
"..., so one has and the third power has as coefficients the triangular numbers 1, 3, 6, 10, 15, 21, ... whose term ""n"" is the binomial coefficient formula_21, so that More generally, for any non-negative integer ""k"" and non-zero real value ""a"", it is true that Note that, since one can find the ordinary generating function for the sequence 0, 1, 4, 9, 16, ... of square numbers by linear combination of binomial-coefficient generating sequences: We may also expand alternately to generate this same sequence of squares as a sum of derivatives of the geometric series in the following"
"210 (number) 210 (two hundred [and] ten) is the natural number following 209 and preceding 211. 210 is a composite number, an abundant number, and the product of the first four prime numbers (2, 3, 5, and 7), and thus a primorial. It is also the least common multiple of these four prime numbers. It is the sum of eight consecutive prime numbers (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210). It is a triangular number (following 190 and preceding 231), a pentagonal number (following 176 and preceding 247), and the"
"sides of the preceding triangle in this series of triangles, and the shorter leg is equal to the difference between the preceding bypassed Fibonacci number and the shorter leg of the preceding triangle. The first triangle in this series has sides of length 5, 4, and 3. Skipping 8, the next triangle has sides of length 13, 12 (5 + 4 + 3), and 5 (8 − 3). Skipping 21, the next triangle has sides of length 34, 30 (13 + 12 + 5), and 16 (21 − 5). This series continues indefinitely. The triangle sides ""a"", ""b"", ""c"" can"
"puzzle ""What is the next number in the sequence 1, 11, 21, 1211, 111221?"" is sometimes referred to as the Cuckoo's Egg, from a description of Morris in Clifford Stoll's book ""The Cuckoo's Egg"". There are many possible variations on the rule used to generate the look-and-say sequence. For example, to form the ""pea pattern"" one reads the previous term and counts all instances of each digit, listed in order of their first appearance, not just those occurring in a consecutive block. Thus, beginning with the seed 1, the pea pattern proceeds 1, 11 (""one 1""), 21 (""two 1s""), 1211"
"6, 7, 8, 8, 10, 13, 15, 16, 20} are {7, 9, 15}. If also required, the zeroth quartile is 3 and the fourth quartile is 20. Consider an ordered population of 11 data values {3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20}. What are the 4-quantiles (the ""quartiles"") of this dataset? So the first, second and third 4-quantiles (the ""quartiles"") of the dataset {3, 6, 7, 8, 8, 9, 10, 13, 15, 16, 20} are {7, 9, 15}. If also required, the zeroth quartile is 3 and the fourth quartile is 20. When one has a"
"of an ""n"" × ""n"" magic square . Summing up the row sums in Floyd's triangle reveals the doubly triangular numbers (triangular numbers with an index that is triangular) 1 = 1 = T(T(1)) <br><br><br> 1 = 6 = T(T(2)) <br> 2 + 3 <br><br><br> 1 <br> 2 + 3 = 21 = T(T(3))<br> 4 + 5 + 6 <br> Floyd's triangle Floyd's triangle is a right-angled triangular array of natural numbers, used in computer science education. It is named after Robert Floyd. It is defined by filling the rows of the triangle with consecutive numbers, starting with a 1"
"of the ""3n-1""th triangular number. Generalized pentagonal numbers are obtained from the formula given above, but with ""n"" taking values in the sequence 0, 1, −1, 2, −2, 3, −3, 4..., producing the sequence: 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, 40, 51, 57, 70, 77, 92, 100, 117, 126, 145, 155, 176, 187, 210, 222, 247, 260, 287, 301, 330, 345, 376, 392, 425, 442, 477, 495, 532, 551, 590, 610, 651, 672, 715, 737, 782, 805, 852, 876, 925, 950, 1001, 1027, 1080, 1107, 1162, 1190, 1247, 1276, 1335... . Generalized pentagonal numbers are important"
"121 (number) 121 (one hundred [and] twenty-one) is the natural number following 120 and preceding 122. One hundred [and] twenty-one is a square and is the sum of three consecutive primes (37 + 41 + 43). There are no squares besides 121 known to be of the form formula_1, where ""p"" is prime (3, in this case). Other such squares must have at least 35 digits. There are only two other squares known to be of the form n! + 1, supporting Brocard's conjecture. Another example of 121 being of the few examples supporting a conjecture is that Fermat conjectured"
"!19 !11 !26 !9 !31 !45 !11 !68 !13 !16 !5 !10 !25 !63 !31 !1 !41 !54 !8 !29 !28 !7 !64 !27 !17 !24 !43 !31 !2 !24 !21 !12 !19 !59 !6 !25 !10 !12 !34 !2 !48 !26 !28 !16 !12 !3 !10 !22 !57 !21 !22 !21 !9 !16 !5 !44 !9 !29 !11 !9 !10 !57 !22 !2 !18 !24 !63 !39 !109 !41 !42 !10 !117 !11 !29 !45 !42 !45 !93 !41 !3 !29 !3 !27 !41 !3 !40 !2 !22 !60 !16 !61 !57 !30 !2 !5 !38 !20 !10 !12"
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"the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. ""p"" is given by the formula: for ""n"" ≥ 1. The first few pentagonal numbers are: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 2035, 2147, 2262, 2380, 2501, 2625, 2752, 2882, 3015, 3151, 3290, 3432, 3577, 3725, 3876, 4030, 4187... . The ""n""th pentagonal number is one third"
"16, 25, 36, 49, ... Sum of Squares 0, 1, 5, 14, 30, 55, 91, 140, 204, 285, ... )number of units in a square-based pyramid) The bimagic series is the 1st, 4th, 9th in this series (divided by 1, 2, 3, n) etc. so values for the rows and columns in order-1, order-2, order-3 Bimagic squares would be 1, 15, 95, 374, 1105, 2701, 5775, 11180, ... The trimagic series would be related in the same way to the hyper-pyramidal sequence of nested cubes. Cubes 0, 1, 8, 27, 64, 125, 216, ... Sum of Cubes 0, 1, 9,"
"11 /> <section begin=Match 12 /> <section begin=Match 21 /> <section begin=Match 22 /> <section begin=Match 3 /> <section begin=Match 4 /> <section begin=Match 13 /> <section begin=Match 14 /> <section begin=Match 23 /> <section begin=Match 24 /> <section begin=Match 5 /> <section begin=Match 6 /> <section begin=Match 15 /> <section begin=Match 16 /> <section begin=Match 25 /> <section begin=Match 26 /> <section begin=Match 7 /> <section begin=Match 8 /> <section begin=Match 17 /> <section begin=Match 18 /> <section begin=Match 27 /> <section begin=Match 28 /> <section begin=Match 9 /> <section begin=Match 10 /> <section begin=Match 19 /> <section begin=Match"
"as π = 3.1415926535897... are catalogued under representative integer sequences such as decimal expansions (here 3, 1, 4, 1, 5, 9, 2, 6, 5, 3, 5, 8, 9, 7, 9, 3, 2, 3, 8, 4, 6, 2, 6, 4, 3, 3, 8, 3, 2, 7, 9, 5, 0, 2, 8, 8, ... ()), binary expansions (here 1, 1, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 0, ... ()), or continued fraction expansions (here 3, 7, 15, 1, 292, 1, 1, 1, 2, 1, 3, 1, 14,"
"Triangular number A triangular number or triangle number counts objects arranged in an equilateral triangle, as in the diagram on the right. The th triangular number is the number of dots in the triangular arrangement with dots on a side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers , starting at the 0th triangular number, is The triangle numbers are given by the following explicit formulas: where formula_2 is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read"
"15 (number) 15 (fifteen) is a number, numeral, and glyph. It is the natural number following 14 and preceding 16. 15 is: There are 15 perfect matchings of the complete graph ""K"" and 15 rooted binary trees with four labeled leaves, both of these being among the types of object counted by double factorials. With only two exceptions, all prime quadruplets enclose a multiple of 15, with 15 itself being enclosed by the quadruplet (11, 13, 17, 19). Because 15 is the product of distinct Fermat primes, 3 and 5, a regular polygon with 15 sides is constructible with compass"
"sequence has been found. For instance, starting with ""n"" = 12, one gets the sequence 12, 6, 3, 10, 5, 16, 8, 4, 2, 1. ""n"" = 19, for example, takes longer to reach 1: 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. The sequence for ""n"" = 27, listed and graphed below, takes 111 steps (41 steps through odd numbers, in large font), climbing to 9232 before descending to 1. Numbers with a total stopping time longer than that of any smaller starting value form a"
"666 (number) 666 (six hundred [and] sixty-six) is the natural number following 665 and preceding 667. Six hundred and sixty-six is called the ""number of the Beast"" in (most manuscripts of) chapter of the Book of Revelation, of the New Testament, and also in popular culture. 666 is the sum of the first 36 natural numbers (formula_1, i.e. ), and thus it is a triangular number. Notice that ; 15 and 21 are also triangular numbers; and . In base 10, 666 is a repdigit (and therefore a palindromic number) and a Smith number. A prime reciprocal magic square based"
"thirteenth place, the value 233, though another manuscript carries it to the next place: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. Fibonacci did not speak about the golden ratio as the limit of the ratio of consecutive numbers in this sequence. In the 19th century, a statue of Fibonacci was constructed and raised in Pisa. Today it is located in the western gallery of the Camposanto, historical cemetery on the Piazza dei Miracoli. There are many mathematical concepts named after Fibonacci because of a connection to the Fibonacci numbers. Examples include the Brahmagupta–Fibonacci"
"1, 3, 21, 55 are the only triangular Fibonacci numbers, which was conjectured by Vern Hoggatt and proved by Luo Ming. With the exceptions of 1, 8 and 144 (""F"" = ""F"", ""F"" and ""F"") every Fibonacci number has a prime factor that is not a factor of any smaller Fibonacci number (Carmichael's theorem). As a result, 8 and 144 (""F"" and ""F"") are the only Fibonacci numbers that are the product of other Fibonacci numbers . The divisibility of Fibonacci numbers by a prime ""p"" is related to the Legendre symbol formula_105 which is evaluated as follows: If ""p"""
"and 15. This was conjectured in 1913 by Indian mathematician Srinivasa Ramanujan, proposed independently in 1943 by the Norwegian mathematician Wilhelm Ljunggren, and proved in 1948 by the Norwegian mathematician Trygve Nagell. The values of ""n"" correspond to the values of ""x"" as:- The problem of finding all numbers of the form 2 − 1 (Mersenne numbers) which are triangular is equivalent: The values of ""b"" are just those of ""n"" − 3, and the corresponding triangular Mersenne numbers (also known as Ramanujan–Nagell numbers) are: for ""x"" = 1, 3, 5, 11 and 181, giving 0, 1, 3, 15, 4095"
"21 (number) 21 (twenty-one) is the natural number following 20 and preceding 22. 21 is: Note that a necessary condition for ""n"" is that for any ""a"" coprime to ""n"", ""a"" and ""n"" - ""a"" must satisfy the condition above, therefore at least one of ""a"" and ""n"" - ""a"" must only have factor 2 and 5. Let formula_3 donate the quantity of the numbers smaller than ""n"" that only have factor 2 and 5 and that are coprime to ""n"", we instantly have formula_4. We can easily see that for sufficiently large ""n"", formula_5, but formula_6, formula_7 as ""n"""
"300 (number) 300 (three hundred) is the natural number following 299 and preceding 301. The number 300 is a triangular number and the sum of a pair of twin primes (149 + 151), as well as the sum of ten consecutive primes (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47). It is palindromic in 3 consecutive bases: 300 = 606 = 454 = 363, and also in bases 13, 19, 24, 29, 49 and 59. Three hundred is: 301 = 7 × 43. 301 is the sum of three"
"!1 !4 !2 !3 !1 !7 !2 !4 !1 !5 !2 !4 !2 !2 !5 !3 !4 !3 !3 !6 !8 !9 !49 !55 !22 !15 !5 !10 !24 !26 !6 !2 !36 !10 !50 !4 !6 !31 !47 !15 !16 !6 !4 !16 !32 !20 !28 !13 !14 !14 !28 !12 !21 !38 !5 !37 !50 !20 !4 !31 !27 !48 !15 !39 !36 !38 !81 !37 !61 !70 !29 !26 !19 !22 !20 !46 !8 !83 !2 !38 !61 !9 !8 !10 !29 !2 !43 !41 !8 !6 !7 !5 !28 !48 !33 !5 !41 !1 !3 !101"
"goes to infinity, thus formula_4 fails to hold for sufficiently large ""n"". In fact, For every ""n"" > 2, we have and so formula_4 fails to hold when ""n"" > 273 (actually, when ""n"" > 33). Just check a few numbers to see that ""n"" = 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 15, 21. 21 appears in the Padovan sequence, preceded by the terms 9, 12, 16 (it is the sum of the first two of these). 21 is: 21 (number) 21 (twenty-one) is the natural number following 20 and preceding 22. 21 is: Note that"
"its values at consecutive integers by using successive differences. This formula closely resembles a recurrence relation that can be used to define the Bell numbers. The sums of each row of the triangle, 1, 3, 10, 37, ..., are the same sequence of first differences appearing in the second-from-right diagonal of the triangle. The ""n""th number in this sequence also counts the number of partitions of ""n"" elements into subsets, where one of the subsets is distinguished from the others; for instance, there are 10 ways of partitioning three items into subsets and then choosing one of the subsets. A"
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"31 is a centered triangular number, the lowest prime centered pentagonal number and a centered decagonal number. For the Steiner tree problem, 31 is the number of possible Steiner topologies for Steiner trees with 4 terminals. At 31, the Mertens function sets a new low of −4, a value which is not subceded until 110. No integer added up to its base 10 digits results in 31, making 31 a self number. 31 is a repdigit in base 5 (111), and base 2 (11111). The numbers 31, 331, 3331, , , , and are all prime. For a time it"
"1 or 4, and progresses in the sequence 1, 4, 1. The last two digits of a star number in base 10 are always 01, 13, 21, 33, 37, 41, 53, 61, 73, 81, or 93. Unique among the star numbers is 35113, since its prime factors (i.e. 13, 37 and 73) are also consecutive star numbers. Geometrically, the ""n""th star number is made up of a central point and 12 copies of the (""n""−1)th triangular number — making it numerically equal to the ""n""th centered dodecagonal number, but differently arranged. Infinitely many star numbers are also triangular numbers, the"
"18, 12, 28, 8, 30, 16, 20, 16, 24, 12, 36, 18, 24, 16, 40, 12, 42, 20, 24, 22, 46, 16, 42, 20, ... (The quotient can be any natural number, and the quotient = 1 if and only if ""n"" is a prime or a Carmichael number (561, 1105, 1729, 2465, 2821, 6601, 8911, 10585, 15841, ... ), the quotient = 2 if and only if ""n"" is in the sequence: 4, 6, 15, 91, 703, 1891, 2701, 11305, 12403, 13981, 18721, ... ) The least number with ""n"" values of ""b"" are (or 0 if no such"
"the period is 1; for ternary it is 2, and so on. In binary, the sequence of cyclic numbers begins: In ternary: In quaternary: In quinary: In senary: In base 7: In octal: In nonary: In base 11: In duodecimal: In base 13: In base 14: In base 15: In hexadecimal: In base 17: In base 18: In base 19: In base 20: In base 21: In base 22: In base 23: In base 24: In base 25: Note that in ternary (""b"" = 3), the case ""p"" = 2 yields 1 as a cyclic number. While single digits may"
". The leftmost and rightmost diagonals of the Bell triangle both contain the sequence 1, 1, 2, 5, 15, 52, ... of the Bell numbers (with the initial element missing in the case of the rightmost diagonal). The next diagonal parallel to the rightmost diagonal gives the sequence of differences of two consecutive Bell numbers, 1, 3, 10, 37, ..., and each subsequent parallel diagonal gives the sequence of differences of previous diagonals. In this way, as observed, this triangle can be interpreted as implementing the Gregory–Newton interpolation formula, which finds the coefficients of a polynomial from the sequence of"