A359387 Primes p such that the smallest prime factor of (2^(p-1)-1)/(3*p) is greater than p.

11, 23, 47, 59, 83, 107, 167, 179, 227, 263, 347, 359, 383, 443, 467, 479, 503, 563, 587, 647, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1847, 1907, 2027, 2039, 2063, 2099, 2207, 2243, 2447, 2459, 2579, 2687, 2699

Offset: 1

Author

Alain Rocchelli, Dec 29 2022

Keywords

• nonn

Comments

This sequence corresponds to the values of p (>7) in A358527 for which p appears in second position in the factorization of 2^(p-1)-1.

All terms are congruent to 11 mod 12, cf. A068231.

It is conjectured that there are infinitely many terms in this sequence, and their estimated asymptotic density n/a(n) ~ C/(log(a(n)))^2 where C is a constant between 0.7 and 0.9.

Examples

			7 is not a term since for p=7, (2^(p-1)-1)/(3*p) = (2^6-1)/(3*7) = 3 and 3 is not greater than 7.
11 is a term since for p=11, (2^(p-1)-1)/(3*p) = (2^10-1)/(3*11) = 31, which is greater than 11.
23 is a term since (2^22-1)/(3*23) = 60787 = 89*683 and 89 is greater than 23.
		

Crossrefs

Cf. A068231, A096060, A358527.

Programs

Mathematica

q[p_] := AllTrue[Range[p], ! PrimeQ[#] || PowerMod[2, p - 1, 3*p*#] > 1 &]; Select[Prime[Range[4, 400]], q] (* Amiram Eldar, Dec 31 2022 *)

PARI

isok(p) = (p%12==11 && isprime(p)) || return(0); forprime(div=5, p-1, if(Mod(2,div)^(p-1)==1, return(0))); 1;