A114447 Indices of 6-almost prime pentagonal numbers.

32, 48, 64, 72, 81, 91, 99, 108, 112, 117, 123, 135, 139, 144, 152, 155, 160, 162, 176, 195, 207, 208, 216, 219, 240, 252, 264, 272, 275, 279, 292, 297, 300, 323, 324, 327, 331, 342, 347, 351, 355, 375, 376, 378, 399, 405, 417, 425, 435, 444, 450, 451, 455, 464

Offset: 1

Author

Jonathan Vos Post, Feb 14 2006

Keywords

• easy
• nonn

Comments

P(2) = 5 is the only prime pentagonal number, all other factor as P(k) = (k/2)*(3*k-1) or k*((3*k-1)/2) and thus have at least 2 prime factors. P(k) is semiprime iff [k prime and (3*k-1)/2 prime] or [k/2 prime and 3*k-1 prime].

Examples

			a(1) = 32 because P(32) = PentagonalNumber(32) = 32*(3*32-1)/2 = 1520 = 2^4 * 5 * 19 is a 6-almost prime.
a(3) = 64 because P(64) = 64*(3*64-1)/2 = 6112 = 2^5 * 191 is a 6-almost prime.
a(15) = 144 because P(144) = 144*(3*144-1)/2 = 31032 = 2^3 * 3^2 * 431 is a 6-almost prime.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Almost Prime.
• Eric Weisstein's World of Mathematics, Pentagonal Number.

Crossrefs

Cf. A000326, A001222, A046306.

Programs

Mathematica

Select[Range[500], PrimeOmega[PolygonalNumber[5, #]] == 6 &] (* Amiram Eldar, Oct 05 2024 *)

Formula

{a(n)} = {k such that A001222(A000326(k)) = 6}.

{a(n)} = {k such that k*(3*k-1)/2 has exactly 6 prime factors}.

{a(n)} = {k such that A000326(k) is an element of A046306}.

Extensions

82 removed by R. J. Mathar, Dec 22 2010

More terms from Amiram Eldar, Oct 05 2024