A114446 - OEIS

%I #7 Nov 21 2013 12:48:46

%S 27,43,96,107,128,147,180,187,203,224,288,312,336,352,360,387,392,395,

%T 400,411,416,475,480,486,491,495,523,539,544,560,572,587,592,600,603,

%U 619,621,627,635,704,729,735,752,763,779,795,800,810,819,840,843,882

%N Indices of 7-almost prime pentagonal numbers.

%C 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].

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number.</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>

%F {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}.

%e a(1) = 27 because P(27) = PentagonalNumber(27) = 27*(3*27-1)/2 = 1080 = 2^3 * 3^3 * 5 is a 7-almost prime.

%e a(2) = 43 because P(43) = 43*(3*43-1)/2 = 2752 = 2^6 * 43 is a 7-almost

%e prime.

%e 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.

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

%Y Cf. A000326, A001222, A046308.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 14 2006

%E More terms from Harvey P. Dale, Jul 16 2011