A114556 Numbers k such that the k-th heptagonal number is 5-almost prime.

7, 16, 23, 30, 32, 36, 42, 45, 54, 69, 78, 79, 80, 84, 88, 90, 93, 95, 100, 102, 104, 112, 115, 117, 140, 143, 151, 153, 165, 170, 174, 176, 184, 186, 191, 200, 203, 210, 213, 228, 232, 234, 245, 250, 259, 271, 273, 282, 287, 296, 306, 308, 310, 311, 318, 319

Offset: 1

Author

Jonathan Vos Post, Feb 15 2006

Keywords

• easy
• nonn

Examples

			a(1) = 7 because Hep(7) = 7*(5*7-3)/2 = 112 = 2^4 * 7 is 5-almost prime [also 112 = Hep(7) = Hep(Hep(2)) is an iterated heptagonal number].
a(2) = 16 because Hep(16) = 16*(5*16-3)/2 = 616 = 2^3 * 7 * 11 is 5-almost prime.
a(3) = 23 because Hep(23) = 23*(5*23-3)/2 = 1288 = 2^3 * 7 * 23.
a(18) = 100 because Hep(100) = 100*(5*100-3)/2 = 24850 = 2 * 5^2 * 7 * 71.
a(21) = 112 because Hep(112) = 112*(5*112-3)/2 = 31192 = 2^3 * 7 * 557 [also 31192 = Hep(112) = Hep(Hep(7)) = Hep(Hep(Hep(2))) is an 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, A014614.

Programs

Mathematica

Select[Range[400], PrimeOmega[# (5 # - 3)/2] == 5 &] (* Giovanni Resta, Jun 14 2016 *)
Select[Range[400],PrimeOmega[PolygonalNumber[7,#]]==5&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Sep 22 2020 *)

Formula

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

Numbers k such that A000566(k) is a term of A014614.

Numbers k such that A001222(A000566(k)) = 5.

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

Extensions

Corrected and extended by Giovanni Resta, Jun 14 2016