%I #13 Dec 10 2017 05:00:05

%S 28,45,153,171,325,4753,7381,29161,56953,65341,166753,354061,5649841,

%T 6060421,6835753,6924781,12708361,19478161,24231241,52035301,56791153,

%U 147258541,186660181,282304441,326081953,520273153,536657941,704531953,784139401,1215121753

%N Triangular numbers having exactly 6 divisors.

%C Intersection of A000217 (triangular numbers) and A030515 (numbers with exactly 6 divisors).

%C This sequence is also the union of

%C (1) numbers of the form p*(2p-1) where p is prime and 2p-1 is the square of a prime (this sequence begins 45, 325, 7381, 65341, 354061, ...),

%C (2) numbers of the form p^2*(2p^2 - 1) where both p and 2p^2 - 1 are prime (this sequence begins 28, 153, 4753, 29161, ...), and

%C (3) numbers of the form p^2*(2p^2 + 1) where both p and 2p^2 + 1 are prime (the only such number is 171).

%H Jon E. Schoenfield, <a href="/A292989/b292989.txt">Table of n, a(n) for n = 1..10000</a>

%e 28 = 2^2 * 7, so it has 6 divisors: {1, 2, 4, 7, 14, 28};

%e 45 = 3^2 * 5, so it has 6 divisors: {1, 3, 5, 9, 15, 45};

%e 171 = 3^2 * 19, so it has 6 divisors: {1, 3, 9, 19, 57, 171}.

%t Select[Array[PolygonalNumber, 10^5], DivisorSigma[0, #] == 6 &] (* _Michael De Vlieger_, Dec 09 2017 *)

%Y Cf. A000217 (triangular numbers), A030515 (numbers with exactly 6 divisors).

%Y Cf. A067756 (primes p such that 2p-1 is the square of a prime), A106483 (primes p such that 2p^2 - 1 is prime).

%Y Cf. A263951.

%K nonn

%O 1,1

%A _Jon E. Schoenfield_, Dec 08 2017