A212918 Numbers whose sum of prime factors (counted with multiplicity) is a pentagonal number (A000326).

1, 5, 6, 35, 42, 50, 57, 60, 64, 72, 81, 85, 102, 121, 124, 182, 188, 201, 232, 260, 261, 267, 308, 312, 351, 440, 452, 495, 519, 528, 594, 645, 649, 688, 735, 741, 774, 784, 805, 854, 861, 875, 882, 901, 966, 1025, 1027, 1045, 1050, 1081, 1105, 1112, 1119

Offset: 1

Author

Jonathan Vos Post, May 31 2012

Keywords

• nonn
• easy

Comments

This is to pentagonal numbers A000326 as A000290 squares are to A212831 numbers whose sum of prime factors is a square (counted with multiplicity) and as A000217 triangular numbers are to A212849 Numbers whose sum of prime factors (counted with multiplicity) is a triangular number.

Examples

			a(3) = 35 because sopfr(35) = sum of prime factors of 35 = 5 + 7 = 12, and  12 is the 3rd pentagonal number.
		

Links

• G. C. Greubel, Table of n, a(n) for n = 1..5000

Crossrefs

Cf. A000326, A001414, A212831, A212849.

Programs

Mathematica

pentagonalQ[n_] := IntegerQ[(1 + Sqrt[1 + 24*n])/6]; pfs[n_] := Module[{p, e}, {p, e} = Transpose[FactorInteger[n]]; Dot[p, e]]; Select[Range[1500], pentagonalQ[pfs[#]] &] (* T. D. Noe, May 31 2012 *)

PARI

sopfr(n) = my(f=factor(n)); sum(k=1, matsize(f)[1], f[k, 1]*f[k, 2]); \\ A001414
isok(n) = ispolygonal(sopfr(n), 5); \\ Michel Marcus, May 02 2018

Formula

{k such that A001414(k) = sopfr(k) is in A000326(j) = j*(3*j-1)/2 for some integer j}.

Extensions

Corrected by T. D. Noe, May 31 2012