A114447 - OEIS

%I #16 Oct 06 2024 09:14:39

%S 32,48,64,72,81,91,99,108,112,117,123,135,139,144,152,155,160,162,176,

%T 195,207,208,216,219,240,252,264,272,275,279,292,297,300,323,324,327,

%U 331,342,347,351,355,375,376,378,399,405,417,425,435,444,450,451,455,464

%N Indices of 6-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 Amiram Eldar, <a href="/A114447/b114447.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F {a(n)} = {k such that A001222(A000326(k)) = 6}.

%F {a(n)} = {k such that k*(3*k-1)/2 has exactly 6 prime factors}.

%F {a(n)} = {k such that A000326(k) is an element of A046306}.

%e a(1) = 32 because P(32) = PentagonalNumber(32) = 32*(3*32-1)/2 = 1520 = 2^4 * 5 * 19 is a 6-almost prime.

%e a(3) = 64 because P(64) = 64*(3*64-1)/2 = 6112 = 2^5 * 191 is a 6-almost prime.

%e a(15) = 144 because P(144) = 144*(3*144-1)/2 = 31032 = 2^3 * 3^2 * 431 is a 6-almost prime.

%t Select[Range[500], PrimeOmega[PolygonalNumber[5, #]] == 6 &] (* _Amiram Eldar_, Oct 05 2024 *)

%Y Cf. A000326, A001222, A046306.

%K easy,nonn

%O 1,1

%A _Jonathan Vos Post_, Feb 14 2006

%E 82 removed by _R. J. Mathar_, Dec 22 2010

%E More terms from _Amiram Eldar_, Oct 05 2024