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%I #20 Oct 06 2024 09:14:19
%S 12,23,25,26,27,30,33,35,39,42,45,46,52,53,58,59,62,65,66,70,75,76,83,
%T 85,93,94,99,111,114,117,118,119,131,133,134,137,145,146,147,154,155,
%U 161,163,167,173,174,175,178,179,183,190,193,195,202,206,209,214,219,222,226,231,233,235,237,239
%N Numbers k such that the k-th hexagonal number is a 4-almost prime.
%C 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 Amiram Eldar, <a href="/A114455/b114455.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%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/HexagonalNumber.html">Hexagonal Number</a>.
%F Numbers k such that hexagonal number A000384(k) is an element of A014613.
%F Numbers k such that A001222(A000384(k)) = 4.
%F Numbers k such that A001222(k*(2*k-1)) = 4.
%e a(1) = 12 because HexagonalNumber(12) = H(12) = 12*(2*12-1) = 276 = 2^2 * 3 * 23 is a 4-almost prime.
%e a(2) = 23 because H(23) = 23*(2*23-1) = 1035 = 3^2 * 5 * 23 is a 4-almost prime.
%e a(3) = 25 because H(25) = 25*(2*25-1) = 1225 = 5^2 * 7^2 is a 4-almost prime.
%t Select[Range[250],PrimeOmega[#(2#-1)]==4&] (* _Harvey P. Dale_, Feb 18 2013 *)
%Y Cf. A000384, A001222, A014613.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Feb 14 2006
%E 40 removed by _R. J. Mathar_, Dec 22 2010