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%I #32 Oct 18 2017 13:44:57
%S 2,3,5,7,31,13,257,73,683,113,331,109,61681,5419,2796203,1613,3033169,
%T 1321,599479,122921,38737,22366891,8831418697,2931542417,22253377,
%U 268501,131071,28059810762433,279073,54410972897,77158673929,145295143558111,2879347902817,10052678938039,616318177,1133836730401,121369
%N a(n) = largest prime q such that q | 2^p - 2 and p - 1 | q - 1, where p = prime(n).
%C First conjecture: a(n) > prime(n) for all n > 6. _Robert Israel_ tested the author's conjecture up to prime(95) = 499. The prime factorizations of the numbers 2^(p-1)-1 for larger p can be checked in available tables, see A005420.
%C Second conjecture: a(n) = gpf(2^prime(n) - 2) for almost all n, in the sense that the set of exceptions {10, 16, 37, 40, ...} has zero natural density.
%C Primes p for which p - 1 does not divide gpf(2^p - 2) - 1 are 29, 53, 157, 173, ...
%e For prime(5) = 11, 2^11-2 = 2*3*11*31 and 11-1 | 31-1, so a(5) = 31.
%Y Cf. A000040, A005420.
%K nonn
%O 1,1
%A _Thomas Ordowski_, Sep 01 2017
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