A334797 Primes q such that p-1 | q-1 or q-1 | p-1 for every prime p | 2^(q-1)-1.

2, 3, 5, 7, 11, 13, 17, 19, 23, 31, 37, 43, 47, 59, 79, 83, 107, 167, 179, 223, 227, 263, 347, 359, 367, 383, 467, 479, 503, 563, 587, 719, 839, 863, 887, 983, 1019, 1187, 1283, 1307, 1319, 1367, 1439, 1487, 1523, 1619, 1823, 1907, 2027, 2039, 2063, 2099, 2207

Offset

1,1

Comments

Are there infinitely many such primes?

Are there only finitely many such primes that are not safe primes?

Is their set {2, 3, 13, 17, 19, 31, 37, 43, 79, 223, 367} complete?

It is assumed that there are infinitely many safe primes (and their estimated asymptotic density ~ 1.32/(log n)^2 converges to the actual value as far as we know), so the answer to the first question is certainly "yes". - M. F. Hasler, Jun 14 2021

Links

Table of n, a(n) for n=1..53.

Mathematica

seqQ[q_] := PrimeQ[q] && Module[{ps = FactorInteger[2^(q - 1) - 1][[;; , 1]]}, AllTrue[ps, Divisible[# - 1, q - 1] || Divisible[q - 1, # - 1] &]]; Select[Range[100], seqQ] (* Amiram Eldar, Jun 09 2020 *)

Prog

(PARI) isok(q) = {if (! isprime(q), return (0)); my(f=factor(2^(q-1)-1)[, 1]~, qq=q-1); for (k=1, #f, my(pp=f[k]-1); if ((qq % pp) && (pp % qq), return(0)); ); return (1); } \\ Michel Marcus, Jun 09 2020
(PARI) is_A334797(n)={isprime(n)&&!foreach(factor(2^n---1)[, 1], p, n%(p-1)&&(p-1)%n&&return)} \\ M. F. Hasler, Jun 14 2021

Crossrefs

A005385 is a proper subset.

Sequence in context: A216286 A086983 A303436 * A229060 A219528 A341660

Adjacent sequences: A334794 A334795 A334796 * A334798 A334799 A334800

Keyword

nonn,hard

Author

Thomas Ordowski, Jun 09 2020

Extensions

a(17)-a(38) from Amiram Eldar, Jun 09 2020

a(39)-a(53) from Daniel Suteu, Jun 19 2020

Status

approved