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

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..33.

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)) = 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.

Crossrefs

Cf. A000326, A001222, A046306.

Sequence in context: A271784 A114416 A046304 * A090052 A163285 A036329

Adjacent sequences: A114444 A114445 A114446 * A114448 A114449 A114450

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 14 2006

Extensions

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

Status

approved