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2.20 Good primes. 2.21 Happy ... Primes for which there is no shorter sub-sequence of the decimal digits that form a prime. There are exactly 26 minimal primes ...
With an aliquot sum of 16, within an aliquot sequence of five composite numbers (26,16,15,9,4,3,1,0) to the Prime in the 3-aliquot tree. 26 is the only integer that is one greater than a square (5 2 + 1) and one less than a cube (3 3 − 1). [2] 26 is a telephone number, specifically, the number of ways of connecting 5 points with pairwise ...
The progressions of numbers that are 0, 3, or 6 mod 9 contain at most one prime number (the number 3); the remaining progressions of numbers that are 2, 4, 5, 7, and 8 mod 9 have infinitely many prime numbers, with similar numbers of primes in each progression.
A good prime is a prime number whose square is greater than the product of any two primes at the same number of positions before and after it in the sequence of primes. That is, good prime satisfies the inequality > + for all 1 ≤ i ≤ n−1, where p k is the kth prime.
Ω(n), the prime omega function, is the number of prime factors of n counted with multiplicity (so it is the sum of all prime factor multiplicities). A prime number has Ω(n) = 1. The first: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37 (sequence A000040 in the OEIS). There are many special types of prime numbers. A composite number has Ω(n) > 1.
Mersenne primes and perfect numbers are two deeply interlinked types of natural numbers in number theory. Mersenne primes, named after the friar Marin Mersenne, are prime numbers that can be expressed as 2 p − 1 for some positive integer p. For example, 3 is a Mersenne prime as it is a prime number and is expressible as 2 2 − 1.
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A prime number q is a strong prime if q + 1 and q − 1 both have some large (around 500 digits) prime factors. For a safe prime q = 2p + 1, the number q − 1 naturally has a large prime factor, namely p, and so a safe prime q meets part of the criteria for being a strong prime.