A117105 Numbers that are the sum of three positive heptagonal numbers (A000566) in at least one way.

3, 9, 15, 20, 21, 26, 32, 36, 37, 42, 43, 48, 53, 54, 57, 59, 63, 69, 70, 74, 75, 80, 83, 86, 89, 90, 91, 95, 96, 100, 102, 106, 107, 111, 114, 116, 117, 120, 122, 123, 126, 128, 131, 133, 137, 143, 144, 147, 148, 149, 150, 153, 154, 156, 162, 163, 164, 165

Offset: 1

Author

Jonathan Vos Post, Apr 18 2006

Keywords

• easy
• nonn

Comments

7 is the only prime heptagonal number. Primes which are sums of two positive heptagonal numbers include: {2, 19, 41, 73, 89, 113, 149, 167, 193, 223, 229, 269, 293, 337, 347, 367, 383, ...}. Primes which are sums of three positive heptagonal numbers include: {3, 37, 43, 53, 59, 83, 89, 107, 131, 137, 149, 163, 167, 173, 191, 197, 211, 227, 241, 251, 257, 263, 271, ...}.

By definition this does not contain any repeated terms. - N. J. A. Sloane, Aug 15 2020

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000566, A000040, A000326, A003679, A064826, A117065.

Programs

Mathematica

With[{nn=10},Select[Union[Total/@Tuples[PolygonalNumber[7,Range[ nn]],3]], #<=PolygonalNumber[7,nn]-2&]] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Aug 16 2020 *)

Formula

{a(n)} = {A000566} + {A000566} + {A000566} = {a*(5*a-3)/2 + b*(5*b-3)/2 + c*(5*c-3)/2} \ {A000566}.

Extensions

Missing 106 and 131 added by Giovanni Resta, Jun 15 2016

Corrected (deleting duplicates) and extended by Harvey P. Dale, Aug 16 2020