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A334797 Primes q such that p-1 | q-1 or q-1 | p-1 for every prime p | 2^(q-1)-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 (list; graph; refs; listen; history; text; internal format)
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
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
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

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Last modified April 24 06:14 EDT 2024. Contains 371918 sequences. (Running on oeis4.)