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

1,2

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.

Links

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

Formula

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000326, A001414, A212831, A212849.

Sequence in context: A151506 A122008 A248254 * A255197 A253192 A036254

Adjacent sequences: A212915 A212916 A212917 * A212919 A212920 A212921

Keyword

nonn,easy

Author

Jonathan Vos Post, May 31 2012

Extensions

Corrected by T. D. Noe, May 31 2012

Status

approved