A114445 - OEIS

A114445

Indices of 5-almost prime pentagonal numbers.

11, 35, 40, 42, 51, 54, 59, 63, 67, 80, 87, 92, 100, 115, 120, 125, 126, 131, 132, 136, 159, 165, 167, 168, 175, 184, 189, 200, 204, 210, 215, 217, 222, 225, 227, 231, 232, 242, 247, 250, 251, 255, 259, 260, 261, 270, 280, 282, 283, 285, 287, 291, 295, 304, 306

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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)) = 5}.

{a(n)} = {k such that k*(3*k-1)/2 has exactly 5 prime factors}.

{a(n)} = {k such that A000326(k) is an element of A014614}.

EXAMPLE

a(1) = 11 because P(11) = PentagonalNumber(11) = 11*(3*11-1)/2 = 176 = 2^4 * 11 is a 4-almost prime (the prime factors need not be distinct).

a(2) = 35 because P(35) = 35*(3*35-1)/2 = 1820 = 2^2 * 5 * 7 * 13 is a 5-almost prime.

a(13) = 100 because P(100) = 100*(3*100-1)/2 = 14950 = 2 * 5^2 * 13 * 23 is a 5-almost prime.