![]() This works with all perfect numbers 2 p−1(2 p − 1) with odd prime p and, in fact, with all numbers of the form 2 m−1(2 m − 1) for odd integer (not necessarily prime) m. ![]() ![]() ![]() This can be reformulated as follows: adding the digits of any even perfect number (except 6), then adding the digits of the resulting number, and repeating this process until a single digit (called the digital root) is obtained, always produces the number 1. With each resulting triangular number T 7 = 28, T 31 = 496, T 127 = 8128 (after subtracting 1 from the perfect number and dividing the result by 9) ending in 3 or 5, the sequence starting with T 2 = 3, T 10 = 55, T 42 = 903, T 2730 = 3727815, . Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself in symbols, σ 1 ( n ) = 2 n The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. Illustration of the perfect number status of the number 6 For the 2012 film, see Perfect Number (film).
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