A114441 Indices of 3-almost prime pentagonal numbers.

3, 7, 8, 9, 17, 18, 20, 21, 22, 23, 25, 26, 28, 30, 31, 37, 44, 49, 50, 61, 62, 65, 66, 69, 71, 74, 76, 78, 79, 85, 89, 93, 97, 98, 113, 116, 121, 122, 129, 130, 133, 137, 141, 146, 148, 151, 154, 157, 158, 161, 164, 166, 170, 173, 174, 178, 185, 186, 188, 190, 193, 194

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) = 3 because P(3) = PentagonalNumber(3) = 3*(3*3 -1)/2 = 12 = 2^2 * 3 is a 3-almost prime.
a(2) = 7 because P(7) = 7*(3*7 -1)/2 = 70 = 2 * 5 * 7 is a 3-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, A014612, A115709.

Programs

Maple

A000326 := proc(n) n*(3*n-1)/2 ; end: isA014612 := proc(n) option remember ; RETURN( numtheory[bigomega](n) = 3) ; end: for n from 1 to 400 do if isA014612(A000326(n)) then printf("%d,",n) ; fi; od: # R. J. Mathar, Jan 27 2009

Mathematica

Select[Range[200], PrimeOmega[PolygonalNumber[5, #]] == 3 &] (* Amiram Eldar, Oct 06 2024 *)

Formula

{a(n)} = {k such that A001222(A000326(k)) = 3}.

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

{a(n)} = {k such that A000326(k) is an element of A014612}.

Extensions

125 removed, 145 replaced with 146 by R. J. Mathar, Jan 27 2009