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

1,1

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.

Links

Table of n, a(n) for n=1..49.

Example

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.

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 *)

Prog

(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;

Crossrefs

Cf. A068231, A096060, A358527.

Sequence in context: A068231 A185005 A073024 * A161897 A282531 A309851

Adjacent sequences: A359384 A359385 A359386 * A359388 A359389 A359390

Keyword

nonn

Author

Alain Rocchelli, Dec 29 2022

Status

approved