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

1,1

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].

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.

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}.

Example

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.

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 *)

Crossrefs

Cf. A000326, A001222, A014612, A115709.

Sequence in context: A342006 A239937 A375019 * A354023 A043046 A030674

Adjacent sequences: A114438 A114439 A114440 * A114442 A114443 A114444

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 14 2006

Extensions

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

Status

approved