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A114441 Indices of 3-almost prime pentagonal numbers. 0
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..62.

Eric Weisstein's World of Mathematics, Pentagonal Number.

Eric Weisstein's World of Mathematics, Almost Prime.

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

CROSSREFS

Cf. A000326, A001222, A014612, A115709.

Sequence in context: A298984 A094551 A239937 * A043046 A030674 A030684

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

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Last modified December 2 16:46 EST 2020. Contains 338877 sequences. (Running on oeis4.)