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A114441
Indices of 3-almost prime pentagonal numbers.
1
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
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
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