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A114443 Indices of 4-almost prime pentagonal numbers.

%I A114443 #24 Feb 16 2025 08:32:59
%S A114443 12,15,16,19,24,33,36,39,45,47,52,55,56,57,60,68,70,77,82,83,84,88,90,
%T A114443 95,102,103,104,105,110,111,114,119,124,127,138,140,142,143,145,150,
%U A114443 153,156,163,169,172,177,179,182,183,191,196,198
%N A114443 Indices of 4-almost prime pentagonal numbers.
%C A114443 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 A114443 Amiram Eldar, <a href="/A114443/b114443.txt">Table of n, a(n) for n = 1..10000</a>
%H A114443 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.
%H A114443 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PentagonalNumber.html">Pentagonal Number</a>.
%F A114443 {a(n)} = {k such that A001222(A000326(k)) = 4}.
%F A114443 {a(n)} = {k such that k*(3*k-1)/2 has exactly 4 prime factors}.
%F A114443 {a(n)} = {k such that A000326(k) is an element of A014613}.
%e A114443 a(1) = 12 because P(12) = A000326(12) = 12*(3*12-1)/2 = 210 = 2 * 3 * 5 * 7 is a 4-almost prime (in fact the primorial prime(4)#).
%e A114443 a(3) = 16 because P(16) = 16*(3*16-1)/2 = 376 = 2^3 * 47 is a 4-almost prime (the prime factors need not be distinct).
%t A114443 Select[Range[400], PrimeOmega[PolygonalNumber[5, #]] == 4 &] (* _Amiram Eldar_, Oct 06 2024 *)
%Y A114443 Cf. A000326, A001222, A014613, A115709.
%K A114443 easy,nonn
%O A114443 1,1
%A A114443 _Jonathan Vos Post_, Feb 14 2006
%E A114443 82 inserted by _R. J. Mathar_, Dec 22 2010