cp's OEIS Frontend

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.

%I A144313 #3 Mar 31 2012 10:28:53
%S A144313 29,71,113,239,281,449,491,659,701,827,911,953,1373,1499,1583,1667,
%T A144313 1709,1877,2003,2087,2129,2213,2339,2423,2549,2591,2633,2801,2843,
%U A144313 2969,3221,3347,3389,3557,3767,3851,4229,4271,4397,4481,4649,4691,4733,5153,5279
%N 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.
%C A144313 Appears to be necessarily a subset of A007528.
%C A144313 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.
%C A144313 It is conjectured:
%C A144313 - that this sequence is infinite;
%C A144313 - 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;
%C A144313 - that the ratio of Mpe in this sequence to those not approaches a nonzero value;
%C A144313 - that a characteristic function f(n) exists which equals 1 iff n is in S.
%C A144313 Contribution from _Reikku Kulon_, Sep 18 2008: (Start)
%C A144313 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.
%C A144313 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.
%C A144313 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.
%C A144313 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)
%Y A144313 Cf. A000040, A007528, A138694, A131500, A096676, A000043, A000668
%Y A144313 Contribution from _Reikku Kulon_, Sep 18 2008: (Start)
%Y A144313 Cf. A071331, A098237 (semiprimes)
%Y A144313 Cf. A140856, A140732, A142076, A142317, A142905 (a, b, c and d all defined) (End)
%K A144313 easy,nonn
%O A144313 1,1
%A A144313 _Reikku Kulon_, Sep 17 2008