A292989 Triangular numbers having exactly 6 divisors.

28, 45, 153, 171, 325, 4753, 7381, 29161, 56953, 65341, 166753, 354061, 5649841, 6060421, 6835753, 6924781, 12708361, 19478161, 24231241, 52035301, 56791153, 147258541, 186660181, 282304441, 326081953, 520273153, 536657941, 704531953, 784139401, 1215121753

Offset: 1

Author

Jon E. Schoenfield, Dec 08 2017

Keywords

• nonn

Comments

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

This sequence is also the union of

(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, ...),

(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

(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).

Examples

			28 = 2^2 * 7, so it has 6 divisors: {1, 2, 4, 7, 14, 28};
45 = 3^2 * 5, so it has 6 divisors: {1, 3, 5, 9, 15, 45};
171 = 3^2 * 19, so it has 6 divisors: {1, 3, 9, 19, 57, 171}.
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..10000

Crossrefs

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

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

Cf. A263951.

Programs

Mathematica

Select[Array[PolygonalNumber, 10^5], DivisorSigma[0, #] == 6 &] (* Michael De Vlieger, Dec 09 2017 *)