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A114447
Indices of 6-almost prime pentagonal numbers.
1
32, 48, 64, 72, 81, 91, 99, 108, 112, 117, 123, 135, 139, 144, 152, 155, 160, 162, 176, 195, 207, 208, 216, 219, 240, 252, 264, 272, 275, 279, 292, 297, 300, 323, 324, 327, 331, 342, 347, 351, 355, 375, 376, 378, 399, 405, 417, 425, 435, 444, 450, 451, 455, 464
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)) = 6}.
{a(n)} = {k such that k*(3*k-1)/2 has exactly 6 prime factors}.
{a(n)} = {k such that A000326(k) is an element of A046306}.
EXAMPLE
a(1) = 32 because P(32) = PentagonalNumber(32) = 32*(3*32-1)/2 = 1520 = 2^4 * 5 * 19 is a 6-almost prime.
a(3) = 64 because P(64) = 64*(3*64-1)/2 = 6112 = 2^5 * 191 is a 6-almost prime.
a(15) = 144 because P(144) = 144*(3*144-1)/2 = 31032 = 2^3 * 3^2 * 431 is a 6-almost prime.
MATHEMATICA
Select[Range[500], PrimeOmega[PolygonalNumber[5, #]] == 6 &] (* Amiram Eldar, Oct 05 2024 *)
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Jonathan Vos Post, Feb 14 2006
EXTENSIONS
82 removed by R. J. Mathar, Dec 22 2010
More terms from Amiram Eldar, Oct 05 2024
STATUS
approved