login
Numbers k such that the k-th heptagonal number is 4-almost prime.
1

%I #19 Feb 16 2025 08:32:59

%S 6,9,12,18,21,31,35,40,44,47,49,50,56,57,65,66,76,91,107,121,125,127,

%T 129,136,138,145,148,152,154,155,163,164,187,196,201,205,212,220,221,

%U 223,226,230,235,236,237,239,242,246,248,260,268,284,289,292,299,309

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

%H 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 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 4-almost prime.

%F Numbers k such that A000566(k) is a term of A014613.

%F Numbers k such that A001222(A000566(k)) = 4.

%F Numbers k such that A001222(k*(5*k-3)/2) = 4.

%e a(1) = 6 because Hep(6) = 6*(5*6-3)/2 = 81 = 3^4 is 4-almost prime.

%e a(2) = 9 because Hep(9) = 9*(5*9-3)/2 = 189 = 3^3 * 7 is 4-almost prime.

%e a(3) = 12 because Hep(12) = 12*(5*12-3)/2 = 342 = 2 * 3^2 * 19 is 4-almost prime.

%e a(4) = 18 because Hep(18) = 18*(5*18-3)/2 = 783 = 3^3 * 29 is 4-almost prime.

%e [also 783 = Hep(18) = Hep(Hep(3)) is the smallest 4-almost prime iterated heptagonal number].

%e 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 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 Select[Range[500],PrimeOmega[(#(5#-3))/2]==4&] (* _Harvey P. Dale_, Aug 04 2016 *)

%Y Cf. A000040, A000566, A001222, A001358, A014613, A099153.

%K easy,nonn,changed

%O 1,1

%A _Jonathan Vos Post_, Feb 15 2006

%E More terms from _Harvey P. Dale_, Aug 04 2016