A114558 - OEIS

%I #27 Feb 16 2025 08:32:59

%S 15,24,27,33,48,51,55,64,71,75,81,99,105,108,111,119,120,123,126,132,

%T 141,147,150,156,160,162,171,175,177,189,198,199,204,208,215,219,222,

%U 224,249,252,258,261,263,264,267,270,272,280,285,291,294,300,304,335

%N Numbers k such that the k-th heptagonal number is 6-almost prime.

%H Amiram Eldar, <a href="/A114558/b114558.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Vincenzo Librandi)

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/AlmostPrime.html">Almost Prime</a>.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeptagonalNumber.html">Heptagonal Number</a>.

%F Numbers k such that Hep(k) = k*(5*k-3)/2 is 6-almost prime.

%F Numbers k such that A000566(k) is a term of A046306.

%F Numbers k such that A001222(A000566(k)) = 6.

%F Numbers k such that A001222(k*(5*k-3)/2) = 6.

%e a(1) = 15 because Hep(15) = 15*(5*15-3)/2 = 540 = 2^2 * 3^3 * 5 is 6-almost prime.

%e a(2) = 24 because Hep(24) = 24*(5*24-3)/2 = 1404 = 2^2 * 3^3 * 13.

%e a(7) = 55 because Hep(55) = 55*(5*55-3)/2 = 7480 = 2^3 * 5 * 11 * 17 is 6-almost prime [also 7480 = Hep(55) = Hep(Hep(5)) is an iterated heptagonal number].

%e a(11) = 81 because Hep(81) = 81*(5*81-3)/2 = 16281 = 3^5 * 67 [also 16281 = Hep(81) = Hep(Hep(6)) is an iterated heptagonal number].

%e a(24) = 156 because Hep(156) = 156*(5*156-3)/2 = 60606 = 2 * 3^2 * 7 * 13 * 37 is 6-almost prime (and a palindrome).

%e a(30) = 189 because Hep(189) = 189*(5*189-3)/2 = 89019 = 3^4 * 7 * 157 is 6-almost prime [also 89019 = Hep(189) = Hep(Hep(9)) is an iterated heptagonal number].

%t Select[Range[400],Total[Transpose[FactorInteger[# (5#-3)/2]][[2]]]==6&] (* _Harvey P. Dale_, May 15 2011 *)

%Y Cf. A000040, A000566, A001222, A001358, A046306.

%K nonn,easy

%O 1,1

%A _Jonathan Vos Post_, Feb 15 2006

%E More terms from _Harvey P. Dale_, May 15 2011