A236128 -id:A236128 - cp's OEIS frontend

A291691 Primes p such that gpf(lpf(2^p - 1) - 1) = p.

2, 3, 5, 7, 11, 13, 23, 29, 37, 43, 47, 53, 73, 79, 83, 97, 113, 131, 151, 173, 179, 181, 191, 197, 211, 223, 233, 239, 251, 263, 277, 281, 283, 307, 317, 337, 353, 359, 367, 383, 397, 419, 431, 439, 443, 457, 461, 463, 467, 487, 491, 499

Offset: 1

Thomas Ordowski, Aug 30 2017

nonn

This sequence has not been proved to be infinite.
The terms p such that 2^p - 1 is a Mersenne prime are 2, 3, 5, 7, and 13.
If p is prime, then gpf(lpf(2^p - 1) - 1) >= p.
Primes q such that gpf(lpf(2^q - 1) - 1) > q are A292237.

			We have gpf(lpf(2^11 - 1) - 1) = gpf(23 - 1) = 11, so 11 is a term.

Charles R Greathouse IV, Table of n, a(n) for n = 1..119

Cf. A000040, A000043, A006530, A016047, A020639, A236128.

Contains A002515.

lpf[n_] := FactorInteger[n][[1, 1]]; gpf[n_] := FactorInteger[n][[-1, 1]]; Select[ Prime@ Range@ 45, gpf[lpf[2^# - 1] - 1] == # &] (* Giovanni Resta, Aug 30 2017 *)

listp(nn) = forprime(p=2, nn, if (vecmax(factor(vecmin(factor(2^p-1)[,1])-1)[,1]) == p, print1(p, ", "));); \\ Michel Marcus, Aug 30 2017

a(17)-a(26) from Michel Marcus, Aug 30 2017
a(27)-a(34) from Giovanni Resta, Aug 30 2017
a(35)-a(52) from Charles R Greathouse IV, Aug 30 2017

A236485 Primes p such that gpf(lpf(m)-1) = gpf(gpf(m)-1), where m = 2^p-1 is Mersenne composite.

Original entry on oeis.org

11, 29, 53

Offset: 1

Thomas Ordowski, Jan 27 2014

Conjecture: finitely many such p and gpf(lpf(m)-1) = gpf(gpf(m)-1) = p.

No more terms found up to p = 1277, 1277 being the first prime for which the complete factorization of 2^p-1 is not currently known (see GIMPS link). - Michel Marcus, Jan 29 2014

GIMPS, Exponent Status

Cf. A003260, A016047, A236128.

isok(p) = isprime(p) && (mc = (2^p-1)) && (mcpf = (factor(2^p-1))[, 1]) && (length(mcpf) > 1) && (gpf = vecmax(mcpf)) && (lpf = vecmin(mcpf)) && (vecmax(factor(gpf-1)[, 1]) == vecmax(factor(lpf-1)[, 1])); \\ Michel Marcus, Jan 27 2014

cp's OEIS Frontend

A291691 Primes p such that gpf(lpf(2^p - 1) - 1) = p.

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