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A114505 Numbers k such that the k-th hexagonal number is a 7-almost prime.

%I A114505 #21 Feb 16 2025 08:32:59
%S A114505 48,64,68,72,80,88,96,104,108,122,140,162,168,188,203,208,216,228,230,
%T A114505 240,243,264,272,280,308,312,324,360,378,380,396,408,410,424,428,438,
%U A114505 440,446,450,473,486,513,518,527,544,564,567,572,578,620,638,662,666,675,689,696
%N A114505 Numbers k such that the k-th hexagonal number is a 7-almost prime.
%C A114505 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 A114505 Amiram Eldar, <a href="/A114505/b114505.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%H A114505 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.
%H A114505 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HexagonalNumber.html">Hexagonal Number</a>.
%F A114505 Numbers k such that hexagonal number A000384(k) is an element of A046308.
%F A114505 Numbers k such that A001222(A000384(k)) = 7.
%F A114505 Numbers k such that A001222(k*(2*k-1)) = 7.
%e A114505 a(1) = 48 because HexagonalNumber(48) = H(48) = 48*(2*48-1) = 4560 = 2^4 * 3 * 5 * 19 is a 7-almost prime.
%e A114505 a(2) = 64 because H(64) = 64*(2*64-1) = 8128 = 2^6 * 127 is a 7-almost prime.
%t A114505 Select[Range[800],PrimeOmega[#(2#-1)]==7&] (* _Harvey P. Dale_, Jul 20 2013 *)
%t A114505 Position[PrimeOmega[PolygonalNumber[6,Range[700]]],7]//Flatten (* _Harvey P. Dale_, Jan 10 2024 *)
%Y A114505 Cf. A000384, A001222, A046308.
%K A114505 easy,nonn
%O A114505 1,1
%A A114505 _Jonathan Vos Post_, Feb 14 2006