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[1] [2] For example, 20 is a primitive abundant number because: The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number. The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8 respectively, so each of these numbers is a deficient number. The first few primitive abundant numbers are:
The aliquot sequence starting with a positive integer k can be defined formally in terms of the sum-of-divisors function σ 1 or the aliquot sum function s in the following way: [1] = = = > = = = If the s n-1 = 0 condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are convergent, the limit of these ...
In computational number theory, the Lucas test is a primality test for a natural number n; it requires that the prime factors of n − 1 be already known. [ 1 ] [ 2 ] It is the basis of the Pratt certificate that gives a concise verification that n is prime.
The only odd practical number is 1, because if is an odd number greater than 2, then 2 cannot be expressed as the sum of distinct divisors of . More strongly, Srinivasan (1948) observes that other than 1 and 2, every practical number is divisible by 4 or 6 (or both).
In 1904, Cipolla showed how to produce an infinite number of pseudoprimes to base a > 1: let A = (a p - 1)/(a - 1) and let B = (a p + 1)/(a + 1), where p is a prime number that does not divide a(a 2 - 1). Then n = AB is composite, and is a pseudoprime to base a. [3] [4] For example, if a = 2 and p = 5, then A = 31, B = 11, and n = 341 is a ...
The number of ways of writing n as a palindromic ordered sum in which no term is 2 is P(n). For example, P(6) = 4, and there are 4 ways to write 6 as a palindromic ordered sum in which no term is 2: 6 ; 3 + 3 ; 1 + 4 + 1 ; 1 + 1 + 1 + 1 + 1 + 1. The number of ways of writing n as an ordered sum in which each term is odd and greater than 1 is ...
In mathematics, the Jacobsthal numbers are an integer sequence named after the German mathematician Ernst Jacobsthal.Like the related Fibonacci numbers, they are a specific type of Lucas sequence (,) for which P = 1, and Q = −2 [1] —and are defined by a similar recurrence relation: in simple terms, the sequence starts with 0 and 1, then each following number is found by adding the number ...
However, the number of Pythagorean primes up to is frequently somewhat smaller than the number of non-Pythagorean primes; this phenomenon is known as Chebyshev's bias. [1] For example, the only values of n {\displaystyle n} up to 600000 for which there are more Pythagorean than non-Pythagorean odd primes less than or equal to n are 26861 and 26862.