A114447 - OEIS

A114447

Indices of 6-almost prime pentagonal numbers.

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

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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)) = 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.