A144325 - OEIS

%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