A292446 Numbers k such that sigma((k + 1) / 2) is a prime q.

3, 7, 17, 31, 49, 127, 577, 1457, 3361, 4801, 6961, 8191, 10081, 15841, 20401, 31249, 34321, 55777, 57121, 59857, 131071, 167041, 171697, 293377, 524287, 559681, 916657, 982801, 1062881, 1104097, 1158241, 1195057, 1367857, 1407841, 1414561, 1468897, 1659841

Offset: 1

Author

Jaroslav Krizek, Sep 16 2017

Keywords

• nonn
• easy

Comments

Corresponding values of primes q are in A062700.

Prime terms are in A292447.

Mersenne primes p = 2^k - 1 (A000668) are terms: sigma((p + 1) / 2) = sigma((2^k - 1 + 1) / 2) = sigma(2^(k - 1)) = 2^k - 1.

This sequence also has terms of the form p^(q-1) where p and q are odd primes, i.e., A002315(1)^2 = 7^2 and A002315(3)^2 = 239^2. Terms that are not squarefree are 49, 55777, 57121, 167041, 2789521, 50060017, ... - Altug Alkan, Oct 02 2017

Examples

			49 is a term because sigma((49 + 1) / 2) = sigma(25) = 31 (prime).
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000203, A000668, A023194, A023195, A062700, A292447, A292448.

Programs

Magma

[n: n in [1..10^8] | IsOdd(n) and IsPrime(SumOfDivisors((n+1) div 2))];

Mathematica

Select[Range[1,166*10^4,2],PrimeQ[DivisorSigma[1,(#+1)/2]]&] (* Harvey P. Dale, Jun 22 2022 *)

PARI

isok(n) = (n%2) && isprime(sigma((n+1)/2)); \\ Michel Marcus, Sep 16 2017

Formula

a(n) = 2*A023194(n) - 1.