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A359387
Primes p such that the smallest prime factor of (2^(p-1)-1)/(3*p) is greater than p.
2
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.
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
KEYWORD
nonn
AUTHOR
Alain Rocchelli, Dec 29 2022
STATUS
approved