A144325 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.

97, 127, 157, 307, 337, 367, 487, 547, 607, 757, 787, 907, 967, 997, 1087, 1117, 1237, 1447, 1567, 1627, 1657, 1747, 1777, 1867, 1987, 2287, 2437, 2617, 2647, 2677, 2767, 2797, 2857, 2887, 3067, 3217, 3307, 3457, 3517, 3547, 3607, 3637, 3727, 3847, 3907

Offset: 1

Author

Reikku Kulon, Sep 17 2008, Sep 18 2008

Keywords

• easy
• nonn

Comments

Every member is congruent to 7 modulo 10.

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.

It is conjectured:

- that this sequence is infinite;

- 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;

- that the ratio of Mpe in S to those not approaches one;

- that a characteristic function f(n) exists which equals 1 iff n is in S;

- 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.

Crossrefs

Cf. A000040, A000043, A000668, A144313, A144315