A281312 Numbers n such that sigma(4*(n-1)) is prime.

2, 5, 17, 1025, 16385, 65537, 268435457, 288230376151711745, 77371252455336267181195265, 20282409603651670423947251286017, 21267647932558653966460912964485513217

Offset: 1

Author

Jaroslav Krizek, Jan 19 2017

Keywords

• nonn

Comments

Conjecture: the next terms are: 288230376151711745, 77371252455336267181195265, 20282409603651670423947251286017, 21267647932558653966460912964485513217.

Conjecture: prime terms are in A258429: 2, 5, 17, 65537.

Conjecture: corresponding primes p are Mersenne primes (A000668) > 3.

Sigma is multiplicative, and sigma(m) > 1 for all m > 1, so sigma(m) can be prime only if m is a prime power. Hence all n in this sequence are of the form 2^m + 1 for some m >= 0. This proves the above conjectures and leads to an explicit formula (q.v.) in terms of the Mersenne numbers. - Charles R Greathouse IV, Mar 01 2017

Crossrefs

Cf. A000203, A000668, A193553, A258429.

Programs

Magma

[n: n in [2..100000] | IsPrime(SumOfDivisors(4*(n-1)))]

PARI

isok(n) = isprime(sigma(4*(n-1))); \\ Michel Marcus, Jan 21 2017

Formula

a(n) = 2^(A000043(n+1)-3) + 1. - Charles R Greathouse IV, Mar 01 2017

Extensions

a(7) = 268435457 confirmed by Jon E. Schoenfield, Jan 20 2017

a(8)-a(11) from Charles R Greathouse IV, Mar 01 2017