A245365 Semiprimes of the form n*(3*n-1)/2.

22, 35, 51, 145, 247, 287, 1247, 1717, 2147, 2501, 3151, 4187, 5017, 7957, 11051, 13207, 15251, 16801, 17767, 20827, 26867, 33227, 49051, 63551, 68587, 71177, 76501, 81317, 96647, 112477, 118301, 128627, 147737, 159251, 182527, 232657, 237407, 241001, 250717

Offset: 1

Author

K. D. Bajpai, Jul 19 2014

Keywords

• nonn

Comments

Semiprimes among pentagonal numbers A000326 = { (3*n^2-n)/2; n >= 0 }.

We can have an odd prime n = 2k + 1 and (3n - 1)/2 = 3k + 1 also prime, i.e., k in A130800, or n = 2p with p prime and 3n - 1 = 6p - 1 also prime, i.e., p in A158015. Considering the ratio of the two prime factors, the two possibilities are mutually exclusive, so this is the disjoint union of {A033570(n)=(2n+1)(3n+1); n in A130800} = A255584 and {p*(6p-1); p in A158015}. - M. F. Hasler, Dec 13 2019

Examples

			n=6: (3*n^2-n)/2 = 51 = 3 * 17 which is semiprime. Hence, 51 appears in the sequence.
n=10: (3*n^2-n)/2 = 145 = 5 * 29 which is semiprime. Hence, 145 appears in the sequence.
		

Links

• K. D. Bajpai, Table of n, a(n) for n = 1..10400

Crossrefs

Cf. A001358, A000326.

Programs

Mathematica

Select[Table[(3*n^2 - n)/2, {n, 500}], PrimeOmega[#] == 2 &]

PARI

select(n->bigomega(n)==2, vector(1000, n, (3*n^2-n)/2)) \\ Colin Barker, Jul 20 2014