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%I #3 Mar 31 2012 10:28:53
%S 97,127,157,307,337,367,487,547,607,757,787,907,967,997,1087,1117,
%T 1237,1447,1567,1627,1657,1747,1777,1867,1987,2287,2437,2617,2647,
%U 2677,2767,2797,2857,2887,3067,3217,3307,3457,3517,3547,3607,3637,3727,3847,3907
%N Prime numbers p such that p - 1 is the third a-figurate number, sixth b-figurate number and possibly twelfth c-figurate number for some a, b and c and not a d-figurate number for any nontrivial d.
%C Every member is congruent to 7 modulo 10.
%C The 12th Mersenne prime exponent (Mpe, A000043) 127 is a member: 126 is the third 43-figurate number and the sixth 10-figurate number and is not a k-figurate number for any other k except 126 (trivially). Several other Mersenne prime exponents are members of this sequence; the next is 607.
%C It is conjectured:
%C - that this sequence is infinite;
%C - that there is a unique set {3, 6, 8, 12, 24, 36, ...} giving the possible orders in k-figurate numbers for the set S of all Mpe for which Mpe - 1 is a (3, ...) k-figurate number;
%C - that the ratio of Mpe in S to those not approaches one;
%C - that a characteristic function f(n) exists which equals 1 iff n is in S;
%C - that all Mersenne primes greater than thirty-one can be characterized by this entry, A144313 and A144315; or by no more than two additional sequences related to (4, 52) and (4, 187) k-figurate numbers.
%Y Cf. A000040, A000043, A000668, A144313, A144315
%K easy,nonn
%O 1,1
%A _Reikku Kulon_, Sep 17 2008, Sep 18 2008
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