A114446 Indices of 7-almost prime pentagonal numbers.

27, 43, 96, 107, 128, 147, 180, 187, 203, 224, 288, 312, 336, 352, 360, 387, 392, 395, 400, 411, 416, 475, 480, 486, 491, 495, 523, 539, 544, 560, 572, 587, 592, 600, 603, 619, 621, 627, 635, 704, 729, 735, 752, 763, 779, 795, 800, 810, 819, 840, 843, 882

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)) = 7}. {a(n)} = {k such that k*(3*k-1)/2 has exactly 7 prime factors}. {a(n)} = {k such that A000326(k) is an element of A046308}.

Example

a(1) = 27 because P(27) = PentagonalNumber(27) = 27*(3*27-1)/2 = 1080 = 2^3 * 3^3 * 5 is a 7-almost prime.
a(2) = 43 because P(43) = 43*(3*43-1)/2 = 2752 = 2^6 * 43 is a 7-almost
prime.
a(7) = 180 because P(180) = 180*(3*180-1)/2 = 48510 = 2 * 3^2 * 5 x 7^2 * 11 is a 7-almost prime.

Mathematica

Select[Range[2000], PrimeOmega[# (3#-1)/2]==7&] (* Harvey P. Dale, Jul 16 2011 *)

Crossrefs

Cf. A000326, A001222, A046308.

Sequence in context: A124940 A272829 A306118 * A141229 A253919 A357077

Adjacent sequences: A114443 A114444 A114445 * A114447 A114448 A114449

Keyword

easy,nonn

Author

Jonathan Vos Post, Feb 14 2006

Extensions

More terms from Harvey P. Dale, Jul 16 2011

Status

approved