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%I #15 Jan 15 2023 14:14:23
%S 8,14,16,18,20,24,28,36,38,40,41,44,54,74,77,78,84,86,90,92,100,102,
%T 105,110,113,123,124,125,126,130,132,135,136,143,148,149,153,156,164,
%U 165,170,171,184,185,186,194,207,210,213,215,218,220,225,232,234,236
%N Numbers n such that the n-th hexagonal number is a 5-almost prime.
%C There are no prime hexagonal numbers. The n-th hexagonal number A000384(n) = n*(2*n-1) is semiprime iff both n and 2*n-1 are prime iff A000384(n) is an element of A001358 iff n is an element of A005382.
%H Harvey P. Dale, <a href="/A114456/b114456.txt">Table of n, a(n) for n = 1..1000</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number.</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/AlmostPrime.html">Almost Prime.</a>
%F n such that hexagonal number A000384(n) is an element of A014614. n such that A001222(A000384(n)) = 5. n such that A001222(n*(2*n-1)) = 5.
%e a(1) = 8 because HexagonalNumber(8) = H(8) = 8*(2*8-1) = 120 = 2^3 * 3 * 5 is a 5-almost prime.
%e a(2) = 14 because H(14) = 14*(2*14-1) = 378 = 2 * 3^3 * 7 is a 5-almost prime.
%e a(3) = 18 because H(18) = 18*(2*18-1) = 630 = 2 * 3^2 * 5 * 7 is a 5-almost prime.
%e a(20) = 100 because H(100) = 100*(2*100-1) = 19900 = 2^2 * 5^2 * 199 is a 5-almost prime.
%t Select[Range[300], PrimeOmega[#*(2*# - 1)] == 5 &] (* _Giovanni Resta_, Jun 14 2016 *)
%t Select[Range[300],PrimeOmega[PolygonalNumber[6,#]]==5&] (* _Harvey P. Dale_, Jan 15 2023 *)
%Y Cf. A000384, A001222, A014614.
%K easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Feb 14 2006
%E Missing a(3)=16 and more terms from _Giovanni Resta_, Jun 14 2016
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