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

6, 9, 12, 18, 21, 31, 35, 40, 44, 47, 49, 50, 56, 57, 65, 66, 76, 91, 107, 121, 125, 127, 129, 136, 138, 145, 148, 152, 154, 155, 163, 164, 187, 196, 201, 205, 212, 220, 221, 223, 226, 230, 235, 236, 237, 239, 242, 246, 248, 260, 268, 284, 289, 292, 299, 309

Offset: 1

Author

Jonathan Vos Post, Feb 15 2006

Keywords

• easy
• nonn

Examples

			a(1) = 6 because Hep(6) = 6*(5*6-3)/2 = 81 = 3^4 is 4-almost prime.
a(2) = 9 because Hep(9) = 9*(5*9-3)/2 = 189 = 3^3 * 7 is 4-almost prime.
a(3) = 12 because Hep(12) = 12*(5*12-3)/2 = 342 = 2 * 3^2 * 19 is 4-almost prime.
a(4) = 18 because Hep(18) = 18*(5*18-3)/2 = 783 = 3^3 * 29 is 4-almost prime.
[also 783 = Hep(18) = Hep(Hep(3)) is the smallest 4-almost prime iterated heptagonal number].
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).
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].
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale)
• Eric Weisstein's World of Mathematics, Almost Prime.
• Eric Weisstein's World of Mathematics, Heptagonal Number.

Crossrefs

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

Programs

Mathematica

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

Formula

Numbers k such that Hep(k) = k*(5*k-3)/2 is 4-almost prime.

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

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

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

Extensions

More terms from Harvey P. Dale, Aug 04 2016