A294074 Primes of the form p*2^k + 1, where p is an odd prime and k is odd.

7, 11, 23, 41, 47, 59, 83, 89, 97, 107, 137, 167, 179, 227, 233, 263, 347, 353, 359, 383, 467, 479, 503, 563, 569, 587, 641, 719, 809, 839, 857, 863, 887, 929, 983, 1019, 1049, 1097, 1187, 1193, 1283, 1307, 1319, 1367, 1409, 1433, 1439, 1487, 1523, 1619, 1697

Offset: 1

Author

Arkadiusz Wesolowski, Feb 07 2018

Keywords

• nonn

Comments

I conjecture that a number of the form p*2^k + 1 (with odd prime p and odd k) belongs to this sequence if and only if p*2^k + 1 divides (p + 2)^(p*2^k) - 1.

This conjecture has been verified for n up to 10^10.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Subsequence of A058500.

Programs

Maple

filter:= proc(n) local k; if not isprime(n) then return false fi; k:= padic:-ordp(n-1,2); k::odd and isprime((n-1)/2^k) end proc:
select(filter, [seq(n,n=3..2000,2)]); # Robert Israel, Mar 13 2018

Mathematica

lst = {}; Do[v = IntegerExponent[m - 1, 2]; If[OddQ[v], If[PrimeQ[(m - 1)/2^v] && PrimeQ[m], AppendTo[lst, m]]], {m, 3, 1697, 2}]; lst

PARI

isok(p) = isprime(p) && (pp=p-1) && (v=valuation(pp,2)) && (v%2) && isprime(pp/2^v); \\ Michel Marcus, Feb 09 2018