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A114456 Numbers k such that the k-th hexagonal number is a 5-almost prime.

%I A114456 #24 Feb 16 2025 08:32:59
%S A114456 8,14,16,18,20,24,28,36,38,40,41,44,54,74,77,78,84,86,90,92,100,102,
%T A114456 105,110,113,123,124,125,126,130,132,135,136,143,148,149,153,156,164,
%U A114456 165,170,171,184,185,186,194,207,210,213,215,218,220,225,232,234,236
%N A114456 Numbers k such that the k-th hexagonal number is a 5-almost prime.
%C A114456 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 A114456 Amiram Eldar, <a href="/A114456/b114456.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%H A114456 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.
%H A114456 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>.
%F A114456 Numbers k such that hexagonal number A000384(k) is an element of A014614.
%F A114456 Numbers k such that A001222(A000384(k)) = 5.
%F A114456 Numbers k such that A001222(k*(2*k-1)) = 5.
%e A114456 a(1) = 8 because HexagonalNumber(8) = H(8) = 8*(2*8-1) = 120 = 2^3 * 3 * 5 is a 5-almost prime.
%e A114456 a(2) = 14 because H(14) = 14*(2*14-1) = 378 = 2 * 3^3 * 7 is a 5-almost prime.
%e A114456 a(3) = 18 because H(18) = 18*(2*18-1) = 630 = 2 * 3^2 * 5 * 7 is a 5-almost prime.
%e A114456 a(20) = 100 because H(100) = 100*(2*100-1) = 19900 = 2^2 * 5^2 * 199 is a 5-almost prime.
%t A114456 Select[Range[300], PrimeOmega[#*(2*# - 1)] == 5 &] (* _Giovanni Resta_, Jun 14 2016 *)
%t A114456 Select[Range[300],PrimeOmega[PolygonalNumber[6,#]]==5&] (* _Harvey P. Dale_, Jan 15 2023 *)
%Y A114456 Cf. A000384, A001222, A014614.
%K A114456 easy,nonn
%O A114456 1,1
%A A114456 _Jonathan Vos Post_, Feb 14 2006
%E A114456 Missing a(3)=16 and more terms from _Giovanni Resta_, Jun 14 2016