A292448 Primes q of the form sigma((p + 1) / 2) where p is a prime.

3, 7, 13, 31, 127, 307, 1723, 2801, 3541, 8191, 19531, 86143, 131071, 492103, 524287, 552793, 684757, 704761, 735307, 797161, 1353733, 1886503, 3413257, 3894703, 5473261, 7094233, 7781311, 9250723, 10378063, 12655807, 18143341, 19443691, 22292563, 23907211

Offset: 1

Author

Jaroslav Krizek, Sep 16 2017

Keywords

• nonn

Comments

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 = p.

2801 is the smallest term of the form 6*k + 5. The next one is 39449441. Note that both of them are of the form 1 + t + t^2 + t^3 + t^4 where t is a prime number. - Altug Alkan, Oct 03 2017

Examples

			Prime 13 is a term because there is prime 17 with sigma((17 + 1) / 2) = sigma(9) = 13.
		

Crossrefs

Cf. A000203, A000668, A292447.

Programs

Magma

m := 5*10^7; Set(Sort([SumOfDivisors((n+1) div 2): n in [1..2*m] | IsPrime(n) and IsPrime(SumOfDivisors((n+1) div 2)) and SumOfDivisors((n+1) div 2) le m])); // corrected by Amiram Eldar, Oct 08 2021

Mathematica

max = 10^6; Select[Union@ Reap[Do[If[PrimeQ@ #, Sow@ #] &@DivisorSigma[1, (Prime@ i + 1)/2], {i, max}] ][[-1, 1]], # < Prime[max]/2 &] (* Michael De Vlieger, Sep 16 2017, corrected by Amiram Eldar, Oct 08 2021 *)

PARI

lista(nn) = {my(list = List()); forprime(p=3, 2*nn, if (isprime(q=sigma((p+1)/2)), listput(list, q));); select(x->(x <= nn), vecsort(Vec(list)));} \\ Michel Marcus, Oct 08 2021