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%I #26 Jan 26 2014 14:25:30
%S 2,3,5,7,11,13,29,53
%N Primes p such that gpf(gpf(2^p-1)-1) = p.
%C 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 20 2014
%C Conjecture: gpf(gpf(2^p-1)-1) = p for finitely many p.
%C Conjecture: gpf(lpf(2^p-1)-1) = p for infinitely many p.
%C Namely, for p = 2, 3, 5, 7, 11, 13, 23, 29, 37, 43, 47, 53, ... - _Michael B. Porter_, Jan 26 2014
%C Note that gpf(lpf(2^p-1)-1) = gpf(gpf(2^p-1)-1) = p for p = 2, 3, 5, 7, 11, 13, 29, 53. See DATA.
%H GIMPS, <a href="http://mersenne.org/report_exponent/">Exponent Status</a>
%e For prime p=2, 2^p-1=3, gpf(3)=3, gpf(3-1)=2, so 2 is in the sequence.
%e For prime p=3, 2^p-1=7, gpf(7)=7, gpf(7-1)=3, so 3 is in the sequence.
%t Select[Prime[Range[25]], FactorInteger[FactorInteger[2^# - 1][[-1, 1]] - 1][[-1, 1]] == # &] (* _Alonso del Arte_, Jan 19 2014 *)
%o (PARI) isok(p) = isprime(p) && (q = (vecmax(factor(2^p-1)[,1]))) && (vecmax(factor(q-1)[,1]) == p); \\ _Michel Marcus_, Jan 19 2014
%Y Cf. A003260, A006530.
%K nonn,more
%O 1,1
%A _Thomas Ordowski_, Jan 19 2014
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