"Sublime number In number theory, a sublime number is a positive integer which has a perfect number of positive factors (including itself), and whose positive factors add up to another perfect number. The number 12, for example, is a sublime number. It has a perfect number of positive factors (6): 1, 2, 3, 4, 6, and 12, and the sum of these is again a perfect number: 1 + 2 + 3 + 4 + 6 + 12 = 28. There are only two known sublime numbers, 12 and (2)(2 − 1)(2 − 1)(2 − 1)(2 − 1)(2 − 1)(2"
"formula_2 is a prime of the form formula_3 for prime formula_4—what is now called a Mersenne prime. Two millenia later, Euler proved that all even perfect numbers are of this form. This is known as the Euclid–Euler theorem. It is not known whether there are any odd perfect numbers, nor whether infinitely many perfect numbers exist. The first perfect number is 6. Its proper divisors are 1, 2, and 3, and 1 + 2 + 3 = 6. Equivalently, the number 6 is equal to half the sum of all its positive divisors: ( 1 + 2 + 3 +"
"6 6 (six) is the natural number following 5 and preceding 7. The SI prefix for 1000 is exa- (E), and for its reciprocal atto- (a). 6 is the smallest positive integer which is neither a square number nor a prime number. Six is the second smallest composite number; its proper divisors are , and . Since six equals the sum of its proper divisors, six is the smallest perfect number, Granville number, and formula_1-perfect number. As a perfect number: Six is the only number that is both the sum and the product of three consecutive positive numbers. Unrelated to"
"Perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. ""σ""(""n"") = 2""n"". This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called (""perfect"", ""ideal"", or ""complete number""). Euclid also proved a formation rule (IX.36) whereby formula_1 is an even perfect number whenever"
"6 being a perfect number, a Golomb ruler of length 6 is a ""perfect ruler"". Six is a congruent number. Six is the first discrete biprime (2 × 3) and the first member of the (2 × ""q"") discrete biprime family. Six is a unitary perfect number, a primary pseudoperfect number, a harmonic divisor number and a superior highly composite number, the last to also be a primorial. The next superior highly composite number is . The next primorial is . There are no Graeco-Latin squares with order 6. If ""n"" is a natural number that is not 2 or"
"6 ) ÷ 2 = 6. The next perfect number is 28: 28 = 1 + 2 + 4 + 7 + 14. This is followed by the perfect numbers 496 and 8128 . In about 300 BC Euclid showed that if 2 − 1 is prime then 2(2 − 1) is perfect. The first four perfect numbers were the only ones known to early Greek mathematics, and the mathematician Nicomachus had noted 8128 as early as 100 AD. Philo of Alexandria in his first-century book ""On the creation"" mentions perfect numbers, claiming that the world was created in 6"
"a prime number. The sum 31 multiplied by 16 (the 5th term in the series) equals 496, which is a perfect number. Book IX, Proposition 35, proves that in a geometric series if the first term is subtracted from the second and last term in the sequence, then as the excess of the second is to the first—so is the excess of the last to all those before it. (This is a restatement of our formula for geometric series from above.) Applying this to the geometric progression 31, 62, 124, 248, 496 (which results from 1, 2, 4, 8, 16"
"numbers. That is, the formula_1-deficient numbers are the natural numbers for that the sum of their divisors in formula_1 is strictly less than themselves: Numbers that fulfill equality in the above definition are known as formula_1-perfect numbers. That is, the formula_1-perfect numbers are the natural numbers that are equal the sum of their divisors in formula_1. The first few formula_1-perfect numbers are: Every perfect number is also formula_1-perfect. However, there are numbers such as 24 which are formula_1-perfect but not perfect. The only known formula_1-perfect number with three distinct prime factors is 126 = 2 · 3 · 7 ."
"64 (number) 64 (sixty-four) is the natural number following 63 and preceding 65. Sixty-four is the square of 8, the cube of 4, and the sixth power of 2. It is the smallest number with exactly seven divisors. It is the lowest positive power of two that is adjacent to neither a Mersenne prime nor a Fermat prime. 64 is the sum of Euler's totient function for the first fourteen integers. It is also a dodecagonal number and a centered triangular number. 64 is also the first whole number that is both a perfect square and a perfect cube. Since"
"number. Most abundant numbers are also semiperfect; abundant numbers which are not semiperfect are called weird numbers. Perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its proper positive divisors, that is, the sum of its positive divisors excluding the number itself (also known as its aliquot sum). Equivalently, a perfect number is a number that is half the sum of all of its positive divisors (including itself) i.e. ""σ""(""n"") = 2""n"". This definition is ancient, appearing as early as Euclid's Elements (VII.22) where it is called (""perfect"", ""ideal"", or"
"part indebted to Philo of Alexandria, who explains (""De Op. Mund"". 3) that it is the first perfect number, ""i.e."", according to Euclid's definition, one equal to the sum of the numbers 1, 2, 3 which divide it without remainder (Aug. ""de Civ. Dei"", xi. 30), the second such number being 28, which is the sum of its divisors 1, 2, 4, 7, 14 (""Orig. t."" 28 in ""S. Joann"".); that being 2 × 3 it arises from the marriage of a male and female, ""i.e."", odd and even number; that there are six directions of motion, forward, backward, right,"
"unknown whether there exists even a single odd perfect number. A perfect number is a natural number that equals the sum of its proper divisors, the numbers that are less than it and divide it evenly. For instance, the proper divisors of 6 are 1, 2, and 3, which sum to 6, so 6 is perfect. A Mersenne prime is a prime number of the form ""M"" = 2 − 1; for a number of this form to be prime, ""p"" itself must also be prime. The Euclid–Euler theorem states that an even natural number is perfect if and only"
"the perfection of 6: Augustine and Alcuin wrote that God had created the world in 6 days because that was the perfect number. The Greek mathematicians had regarded as perfect that number which equals the sum of its divisors that are smaller than itself. Such a number is neither 3 nor 7 nor 10, but 6, for 1 + 2 + 3 = 6. But there are more numbers that show this property, such as 28, which = 1 + 2 + 4 + 7 + 14. It became customary to call such numbers ""perfect."" Euclid gave a formula for"
"The number 10 was thought perfect because there are 10 fingers to the two hands. The number 6 was believed perfect for being divisible in a special way: a sixth part of that number constitutes unity; a third is two; a half — three; two-thirds () is four; five-sixths (""pentamoiron"") is five; six is the perfect whole. The ancients also considered 6 a perfect number because the human foot constituted one-sixth the height of a man, hence the number 6 determined the height of the human body. Thus both numbers, 6 and 10, were credited with perfection, both on purely"
"Almost perfect number In mathematics, an almost perfect number (sometimes also called slightly defective or least deficient number) is a natural number ""n"" such that the sum of all divisors of ""n"" (the sum-of-divisors function ""σ""(""n"")) is equal to 2""n"" − 1, the sum of all proper divisors of ""n"", ""s""(""n"") = ""σ""(""n"") − ""n"", then being equal to ""n"" − 1. The only known almost perfect numbers are powers of 2 with non-negative exponents . Therefore the only known odd almost perfect number is 2 = 1, and the only known even almost perfect numbers are those of the"
"odd unitary perfect number must have only one distinct prime factor, and it is not hard to show that a power of prime cannot be a unitary perfect number, since there are not enough divisors. It is not known whether or not there are infinitely many unitary perfect numbers, or indeed whether there are any further examples beyond the five already known. A sixth such number would have at least nine odd prime factors. Unitary perfect number A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. (A"
"Amicable numbers Amicable numbers are two different numbers so related that the sum of the proper divisors of each is equal to the other number. (A proper divisor of a number is a positive factor of that number other than the number itself. For example, the proper divisors of 6 are 1, 2, and 3.) A pair of amicable numbers constitutes an aliquot sequence of period 2. A related concept is that of a perfect number, which is a number that equals the sum of ""its own"" proper divisors, in other words a number which forms an aliquot sequence of"
"121 = 11. It is the eighth hexagonal number and the fifteenth triangular number, as well as the sum of the first eight triangular numbers, making it also a tetrahedral number. 120 is divisible by the first 5 triangular numbers and the first 4 tetrahedral numbers. 120 is the first multiply perfect number of order three (""a 3-perfect"" or ""triperfect number""). The sum of its factors (including one and itself) sum to 360; exactly three times 120. Note that perfect numbers are order two (""2-perfect"") by the same definition. 120 is divisible by the number of primes below it, 30"
"60 (number) 60 (sixty) () is the natural number following 59 and preceding 61. Being three times 20, it is called ""three score"" in older literature. It is a composite number, with divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, making it a highly composite number. Because it is the sum of its unitary divisors (excluding itself), it is a unitary perfect number, and it is an abundant number with an abundance of 48. Being ten times a perfect number, it is a semiperfect number. It is the smallest number divisible by the numbers"
"listed a few more which are now known to be incorrect. In a manuscript written between 1456 and 1461, an unknown mathematician recorded the earliest European reference to a fifth perfect number, with 33,550,336 being correctly identified for the first time. In 1588, the Italian mathematician Pietro Cataldi also identified the sixth (8,589,869,056) and the seventh (137,438,691,328) perfect numbers, and also proved that every perfect number obtained from Euclid's rule ends with a 6 or an 8. Euclid proved that 2(2 − 1) is an even perfect number whenever 2 − 1 is prime (Elements, Prop. IX.36). For example, the"
"five, and only known, unitary perfect numbers are: 6, 60, 90, 87360, 146361946186458562560000 The respective sums of proper unitary divisors: There are no odd unitary perfect numbers. This follows since one has 2 dividing the sum of the unitary divisors of an odd number (where ""d""*(""n"") is the number of distinct prime divisors of n). One gets this because the sum of all the unitary divisors is a multiplicative function and one has the sum of the unitary divisors of a power of a prime ""p"" is ""p"" + 1 which is even for all odd primes ""p"". Therefore, an"
"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"
"divisors, 36 is a highly composite number. Adding up some subsets of its divisors (e.g., 6, 12, and 18) gives 36, hence 36 is a semiperfect number. This number is the sum of a twin prime pair (17 + 19), the sum of the cubes of the first three positive integers, and also the product of the squares of the first three positive integers. 36 is the number of degrees in the interior angle of each tip of a regular pentagram. The thirty-six officers problem is a mathematical puzzle. The number of possible outcomes (not summed) in the roll of"
"(and only known) 3-hyperperfect number. 326 = 2 × 163. 326 is a nontotient, noncototient, and an untouchable number. 326 is the sum of the 14 consecutive primes (3 + 5 + 7 + 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 + 47). 327 = 3 × 109. 327 is a perfect totient number. 328 = 2 × 41. 328 is a refactorable number, and it is the sum of the first fifteen primes (2 + 3 + 5 + 7 + 11 + 13 + 17 +"
"own image with His length of sixty cubits.."" It is: 60 (number) 60 (sixty) () is the natural number following 59 and preceding 61. Being three times 20, it is called ""three score"" in older literature. It is a composite number, with divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30, and 60, making it a highly composite number. Because it is the sum of its unitary divisors (excluding itself), it is a unitary perfect number, and it is an abundant number with an abundance of 48. Being ten times a perfect number, it is a semiperfect"
"(even) ""perfect"" numbers: where ""p"" and 2 − 1 are prime numbers. Euclid had listed the first four perfect numbers: 6; 28; 496; and 8128. A manuscript of 1456 gave the fifth perfect number: 33,550,336. Gradually mathematicians found further perfect numbers (which are very rare). In 1652 the Polish polymath Jan Brożek noted that there was no perfect number between 10 and 10. Despite over 2,000 years of study, it still is not known whether there exist infinitely many perfect numbers; or whether there are any odd ones. Today the term ""perfect number"" is merely historic in nature, used for"
"6th power, 4th power, cube and square, since 6, 4, 3 and 2 divide 12). In 2002 Romanian mathematician Preda Mihăilescu proved that the only pair of consecutive perfect powers is 2 = 8 and 3 = 9, thus proving Catalan's conjecture. Pillai's conjecture states that for any given positive integer ""k"" there are only a finite number of pairs of perfect powers whose difference is ""k"". This is an unsolved problem. As an alternate way to calculate perfect powers, the recursive approach has yet to be found useful. It is based on the observation that the difference between and"
"(""In Nicomachi arithmeticam"", Leipzig, 1894) states that the Pythagoreans had called the number 6 ""marriage,"" ""health,"" and ""beauty,"" on account of the harmony and accord of that number. The perfect numbers early on came to be treated as the measure of other numbers: those in which the sum of the divisors is greater than the number itself, as in 12, have — since as early as Theon of Smyrna, ca. 130 A.D. — been called ""redundant"" (), ""more than perfect"" (""plus quam perfecti""), or ""abundant numbers"", and those the sum of whose divisors is smaller, as in 8, have been"
"86 (number) 86 (eighty-six) is the natural number following 85 and preceding 87. 86 is: It appears in the Padovan sequence, preceded by the terms 37, 49, 65 (it is the sum of the first two of these). It is conjectured that 86 is the largest n for which the decimal expansion of 2 contains no 0. 86 = (8 × 6 = 48) + (4 × 8 = 32) + (3 × 2 = 6). That is, 86 is equal to the sum of the numbers formed in calculating its multiplicative persistence. To 86 something means ""be out of"""
"not powers of two. The geometric progression 1, 2, 4, 8, 16, 32, … (or, in the binary numeral system, 1, 10, 100, 1000, 10000, 100000, … ) is important in number theory. Book IX, Proposition 36 of ""Elements"" proves that if the sum of the first terms of this progression is a prime number (and thus is a Mersenne prime as mentioned above), then this sum times the th term is a perfect number. For example, the sum of the first 5 terms of the series 1 + 2 + 4 + 8 + 16 = 31, which is"
"Unitary perfect number A unitary perfect number is an integer which is the sum of its positive proper unitary divisors, not including the number itself. (A divisor ""d"" of a number ""n"" is a unitary divisor if ""d"" and ""n""/""d"" share no common factors.) Some perfect numbers are not unitary perfect numbers, and some unitary perfect numbers are not regular perfect numbers. 60 is a unitary perfect number, because 1, 3, 4, 5, 12, 15, and 20 are its proper unitary divisors, and 1 + 3 + 4 + 5 + 12 + 15 + 20 = 60. The first"
"difference between 9 and any higher odd perfect square is eight times a triangular number minus eight. Since all triangular numbers have an odd factor, but no two values of differ by an amount containing an odd factor, the only perfect square of the form is 1, and the only perfect square of the form is 9. Square number In mathematics, a square number or perfect square is an integer that is the square of an integer; in other words, it is the product of some integer with itself. For example, 9 is a square number, since it can be"
"496 (number) 496 (four hundred [and] ninety-six) is the natural number following 495 and preceding 497. 496 is most notable for being a perfect number, and one of the earliest numbers to be recognized as such. As a perfect number, it is tied to the Mersenne prime 31, 2 − 1, with 2 (2 − 1) yielding 496. Also related to its being a perfect number, 496 is a harmonic divisor number, since the number of proper divisors of 496 divided by the sum of the reciprocals of its divisors, 1, 2, 4, 8, 16, 31, 62, 124, 248 and"