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

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.

Mathematica

Select[Range[500], PrimeOmega[PolygonalNumber[5, #]] == 6 &] (* Amiram Eldar, Oct 05 2024 *)

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

More terms from Amiram Eldar, Oct 05 2024

Status

approved