A114443 Indices of 4-almost prime pentagonal numbers.

12, 15, 16, 19, 24, 33, 36, 39, 45, 47, 52, 55, 56, 57, 60, 68, 70, 77, 82, 83, 84, 88, 90, 95, 102, 103, 104, 105, 110, 111, 114, 119, 124, 127, 138, 140, 142, 143, 145, 150, 153, 156, 163, 169, 172, 177, 179, 182, 183, 191, 196, 198

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..52.

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)) = 4}. {a(n)} = {k such that k*(3*k-1)/2 has exactly 4 prime factors}. {a(n)} = {k such that A000326(k) is an element of A014613}.

Example

a(1) = 12 because P(12) = PentagonalNumber(12) = 12*(3*12-1)/2 = 210 = 2 * 3 * 5 * 7 is a 4-almost prime In fact the primorial 4#).
a(3) = 16 because P(16) = 16*(3*16-1)/2 = 376 = 2^3 * 47 is a 4-almost
prime (the prime factors need not be distinct).

Crossrefs

Cf. A000326, A001222, A014613, A115709.

Sequence in context: A115402 A297790 A260462 * A368996 A368995 A188766

Adjacent sequences: A114440 A114441 A114442 * A114444 A114445 A114446

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 14 2006

Extensions

82 inserted by R. J. Mathar, Dec 22 2010

Status

approved