cp's OEIS Frontend

All terms are congruent to 5 modulo 6: A144325(n) and A144313(n) are each congruent to 5 and A144315(n) is congruent to 1.

None of the given terms have more than three distinct prime factors and most have only two. Several are primes.

The multiples of five are all fifth figurate numbers corresponding to polygons having a number of sides k = floor(a(n) / 10) + 2. 3725 = 5 * 5 * 149 is also the 50th pentagonal number. The rest are not figurates, except for 15113 = 7 * 17 * 127, which is the seventeenth 113-figurate number.

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.

Obviously true for the initial terms!

Conjecture: 191, 1217, 1559 and 1901 are not in fact members of this sequence, noting that they are (4, 19) k-figurate numbers; 19 is a member of A138694. Determining whether a Mersenne prime exponent one greater than a (4, 19) k-figurate number exists is sufficient to determine whether these primes are members.