cp's OEIS Frontend

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

%I A359387 #45 Jan 21 2023 02:40:31
%S A359387 11,23,47,59,83,107,167,179,227,263,347,359,383,443,467,479,503,563,
%T A359387 587,647,719,839,863,887,983,1019,1187,1283,1307,1319,1367,1439,1487,
%U A359387 1523,1619,1823,1847,1907,2027,2039,2063,2099,2207,2243,2447,2459,2579,2687,2699
%N A359387 Primes p such that the smallest prime factor of (2^(p-1)-1)/(3*p) is greater than p.
%C A359387 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.
%C A359387 All terms are congruent to 11 mod 12, cf. A068231.
%C A359387 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.
%e A359387 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.
%e A359387 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.
%e A359387 23 is a term since (2^22-1)/(3*23) = 60787 = 89*683 and 89 is greater than 23.
%t A359387 q[p_] := AllTrue[Range[p], ! PrimeQ[#] || PowerMod[2, p - 1, 3*p*#] > 1 &]; Select[Prime[Range[4, 400]], q] (* _Amiram Eldar_, Dec 31 2022 *)
%o A359387 (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;
%Y A359387 Cf. A068231, A096060, A358527.
%K A359387 nonn
%O A359387 1,1
%A A359387 _Alain Rocchelli_, Dec 29 2022