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A114504 Numbers k such that the k-th hexagonal number is a 6-almost prime.

%I A114504 #19 Feb 16 2025 08:32:59
%S A114504 50,56,60,63,81,95,98,112,116,120,138,150,152,158,172,180,182,189,196,
%T A114504 198,204,212,221,238,242,252,260,266,275,276,296,300,304,306,315,328,
%U A114504 332,333,340,344,348,350,356,363,374,375,388,390,405,413,420,423,434,452,455,456,459,462,472
%N A114504 Numbers k such that the k-th hexagonal number is a 6-almost prime.
%C A114504 There are no prime hexagonal numbers. The k-th hexagonal number A000384(k) = k*(2*k-1) is semiprime iff both k and 2*k-1 are primes iff A000384(k) is an element of A001358 iff k is an element of A005382.
%H A114504 Amiram Eldar, <a href="/A114504/b114504.txt">Table of n, a(n) for n = 1..10000</a>
%H A114504 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.
%H A114504 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>.
%F A114504 Numbers k such that hexagonal number A000384(k) is an element of A046306.
%F A114504 Numbers k such that A001222(A000384(k)) = 6.
%F A114504 Numbers k such that A001222(k*(2*k-1)) = 6.
%e A114504 a(1) = 50 because HexagonalNumber(50) = H(50) = 50*(2*50-1) = 4950 = 2 * 3^2 * 5^2 * 11 is a 6-almost prime.
%e A114504 a(2) = 56 because H(56) = 56*(2*56-1) = 6216 = 2^3 * 3 * 7 * 37 is a 6-almost prime.
%e A114504 a(5) = 81 because H(81) = 81*(2*81-1) = 13041 = 3^4 * 7 * 23 is a 6-almost prime.
%t A114504 Select[Range[500], PrimeOmega[PolygonalNumber[6, #]] == 6 &] (* _Amiram Eldar_, Oct 06 2024 *)
%Y A114504 Cf. A000384, A001222, A046306.
%K A114504 easy,nonn
%O A114504 1,1
%A A114504 _Jonathan Vos Post_, Feb 14 2006
%E A114504 199 replaced by 198 by _R. J. Mathar_, Dec 22 2010