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 A145731 Integers n such that for all prime numbers p > 7 there exists an n for which A144912(2, p^n) = 0 iff 2^p - 1 is composite. 0

%I

%S 4,10,19,23,24,25,27,28,30,31,32,33,34,42,48,51,52,53,55,59,60,61,62,

%T 68,69,71,72,74,75,76,77,78,79,80,81,82,84,85,86,91,92,93,95,96,98,99,

%U 100,101,102,103,104,105,106,109,110,112,113,115,116,117,118,119,120,121

%N Integers n such that for all prime numbers p > 7 there exists an n for which A144912(2, p^n) = 0 iff 2^p - 1 is composite.

%C An integer n is excluded from the sequence iff A144912(2, p^n) = 0 for some Mersenne prime exponent p > 7.

%C The given terms are sufficient to identify the Mersenne prime exponents 13, 17, 19 and 31 without error, followed by the incorrect 41 and 59, correct 61, incorrect 71 and correct 89. Additional terms quickly reduce the number of false positives such that, for example, the first thirty Mersenne primes can be identified within minutes using unexceptional software and hardware and, in particular, without primality testing of integers larger than 132049.

%C Noting that A144912(2, k) is a function of k in base 2, it is expected that extremely efficient methods can be found for producing Mersenne primes and perfect numbers within seconds.

%Y Cf. A000040, A000043, A000396, A000668

%K easy,nonn

%O 1,1

%A _Reikku Kulon_, Oct 17 2008

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