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A114554 Numbers k such that the k-th heptagonal number is 4-almost prime.

%I A114554 #19 Feb 16 2025 08:32:59
%S A114554 6,9,12,18,21,31,35,40,44,47,49,50,56,57,65,66,76,91,107,121,125,127,
%T A114554 129,136,138,145,148,152,154,155,163,164,187,196,201,205,212,220,221,
%U A114554 223,226,230,235,236,237,239,242,246,248,260,268,284,289,292,299,309
%N A114554 Numbers k such that the k-th heptagonal number is 4-almost prime.
%H A114554 Amiram Eldar, <a href="/A114554/b114554.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Harvey P. Dale)
%H A114554 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.
%H A114554 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeptagonalNumber.html">Heptagonal Number</a>.
%F A114554 Numbers k such that Hep(k) = k*(5*k-3)/2 is 4-almost prime.
%F A114554 Numbers k such that A000566(k) is a term of A014613.
%F A114554 Numbers k such that A001222(A000566(k)) = 4.
%F A114554 Numbers k such that A001222(k*(5*k-3)/2) = 4.
%e A114554 a(1) = 6 because Hep(6) = 6*(5*6-3)/2 = 81 = 3^4 is 4-almost prime.
%e A114554 a(2) = 9 because Hep(9) = 9*(5*9-3)/2 = 189 = 3^3 * 7 is 4-almost prime.
%e A114554 a(3) = 12 because Hep(12) = 12*(5*12-3)/2 = 342 = 2 * 3^2 * 19 is 4-almost prime.
%e A114554 a(4) = 18 because Hep(18) = 18*(5*18-3)/2 = 783 = 3^3 * 29 is 4-almost prime.
%e A114554 [also 783 = Hep(18) = Hep(Hep(3)) is the smallest 4-almost prime iterated heptagonal number].
%e A114554 a(11) = 49 because Hep(49) = 49*(5*49-3)/2 = 5929 = 7^2 * 11^2 is 4-almost prime (and the smallest such square heptagonal number A046196).
%e A114554 a(27) = 148 because Hep(148) = 148*(5*148-3)/2 = 54538 = 2 * 11 * 37 * 67 is 4-almost prime [also 54538 = Hep(148) = Hep(Hep(8)) is the second smallest 4-almost prime iterated heptagonal number].
%t A114554 Select[Range[500],PrimeOmega[(#(5#-3))/2]==4&] (* _Harvey P. Dale_, Aug 04 2016 *)
%Y A114554 Cf. A000040, A000566, A001222, A001358, A014613, A099153.
%K A114554 easy,nonn
%O A114554 1,1
%A A114554 _Jonathan Vos Post_, Feb 15 2006
%E A114554 More terms from _Harvey P. Dale_, Aug 04 2016