A114443 Indices of 4-almost prime pentagonal numbers.

12, 15, 16, 19, 24, 33, 36, 39, 45, 47, 52, 55, 56, 57, 60, 68, 70, 77, 82, 83, 84, 88, 90, 95, 102, 103, 104, 105, 110, 111, 114, 119, 124, 127, 138, 140, 142, 143, 145, 150, 153, 156, 163, 169, 172, 177, 179, 182, 183, 191, 196, 198

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) = 12 because P(12) = A000326(12) = 12*(3*12-1)/2 = 210 = 2 * 3 * 5 * 7 is a 4-almost prime (in fact the primorial prime(4)#).
a(3) = 16 because P(16) = 16*(3*16-1)/2 = 376 = 2^3 * 47 is a 4-almost prime (the prime factors need not be distinct).
		

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, A014613, A115709.

Programs

Mathematica

Select[Range[400], PrimeOmega[PolygonalNumber[5, #]] == 4 &] (* Amiram Eldar, Oct 06 2024 *)

Formula

{a(n)} = {k such that A001222(A000326(k)) = 4}.

{a(n)} = {k such that k*(3*k-1)/2 has exactly 4 prime factors}.

{a(n)} = {k such that A000326(k) is an element of A014613}.

Extensions

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