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

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..25.

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)) = 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.

Crossrefs

Cf. A000326, A001222, A014614.

Sequence in context: A050287 A096762 A184207 * A054475 A029540 A319971

Adjacent sequences: A114442 A114443 A114444 * A114446 A114447 A114448

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 14 2006

Status

approved