%I #19 Sep 08 2022 08:46:18
%S 2,5,17,1025,16385,65537,268435457,288230376151711745,
%T 77371252455336267181195265,20282409603651670423947251286017,
%U 21267647932558653966460912964485513217
%N Numbers n such that sigma(4*(n-1)) is prime.
%C Conjecture: the next terms are: 288230376151711745, 77371252455336267181195265, 20282409603651670423947251286017, 21267647932558653966460912964485513217.
%C Conjecture: prime terms are in A258429: 2, 5, 17, 65537.
%C Conjecture: corresponding primes p are Mersenne primes (A000668) > 3.
%C 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
%F a(n) = 2^(A000043(n+1)-3) + 1. - _Charles R Greathouse IV_, Mar 01 2017
%o (Magma) [n: n in [2..100000] | IsPrime(SumOfDivisors(4*(n-1)))]
%o (PARI) isok(n) = isprime(sigma(4*(n-1))); \\ _Michel Marcus_, Jan 21 2017
%Y Cf. A000203, A000668, A193553, A258429.
%K nonn
%O 1,1
%A _Jaroslav Krizek_, Jan 19 2017
%E a(7) = 268435457 confirmed by _Jon E. Schoenfield_, Jan 20 2017
%E a(8)-a(11) from _Charles R Greathouse IV_, Mar 01 2017