A144313 Prime numbers p such that p - 1 is the fourth a-figurate number, seventh b-figurate number and possibly tenth c-figurate number for some a, b and c and not a d-figurate number for any nontrivial d.

29, 71, 113, 239, 281, 449, 491, 659, 701, 827, 911, 953, 1373, 1499, 1583, 1667, 1709, 1877, 2003, 2087, 2129, 2213, 2339, 2423, 2549, 2591, 2633, 2801, 2843, 2969, 3221, 3347, 3389, 3557, 3767, 3851, 4229, 4271, 4397, 4481, 4649, 4691, 4733, 5153, 5279

Offset: 1

Author

Reikku Kulon, Sep 17 2008

Keywords

• easy
• nonn

Comments

Appears to be necessarily a subset of A007528.

The 46th Mersenne prime exponent (Mpe, A000043) 43112609 is a member: 43112608 is the fourth 7185436-figurate number and the seventh 2052983-figurate number and is not a k-figurate number for any other k except 43112608 (trivially). Several other Mersenne prime exponents are members of this sequence.

It is conjectured:

- that this sequence is infinite;

- that there is a unique set {4, 7, 10, 16, ...} (A138694?) giving the possible orders in k-figurate numbers for the set S of all Mpe for which Mpe - 1 is a (4, 7) or (4, 10) k-figurate number;

- that the ratio of Mpe in this sequence to those not approaches a nonzero value;

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

Contribution from Reikku Kulon, Sep 18 2008: (Start)

Subset of the integers n such that n is congruent to 29 modulo 42. The case where p - 1 is a tenth c-figurate number occurs when p is also congruent to 281 modulo 630.

The first three primes where c is defined are 281, 911 and 2801, with c = 8, 22, 64; c is congruent to 8 modulo 14. All such primes are necessarily congruent to 1 modulo 10.

The first invalid values of c are 36 and 50, which correspond to the semiprimes 1541 = 23 * 67 and 2171 = 13 * 167. Both of these are members of A071331 and A098237. The next invalid value of c, 78, corresponds to 3431 = 47 * 73, once again a member of both sequences.

The first primes where a, b, c and d are all defined (which therefore excludes them from this sequence) are the consecutive 6581, 7211 and 7841, all members of A140856, A140732, A142076, A142317 and A142905. (End)

Crossrefs

Cf. A000040, A007528, A138694, A131500, A096676, A000043, A000668

Contribution from Reikku Kulon, Sep 18 2008: (Start)

Cf. A071331, A098237 (semiprimes)

Cf. A140856, A140732, A142076, A142317, A142905 (a, b, c and d all defined) (End)