A212849 Numbers whose sum of prime factors (counted with multiplicity) is a triangular number.

1, 3, 8, 9, 21, 25, 26, 30, 32, 36, 38, 44, 68, 86, 105, 106, 112, 115, 125, 126, 138, 150, 155, 160, 164, 178, 180, 186, 187, 192, 195, 203, 206, 208, 216, 231, 234, 243, 266, 275, 290, 299, 302, 305, 323, 330, 338, 343, 348, 352, 365, 366, 380, 396, 404, 413

Offset

1,2

Comments

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

Links

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

Formula

{k such that A001414(k) = sopfr(k) is in A000217}.

Example

sopfr(21) = sum of primes dividing 21 (with repetition) = 10, which is the  4th triangular number, so 21 is in this sequence.
The number 1 is here because the sum of its prime factors is 0, which is a triangular number.

Mathematica

triangularQ[n_] := IntegerQ[Sqrt[1 + 8*n]]; fQ[n_] := Module[{p, e}, {p, e} = Transpose[FactorInteger[n]]; triangularQ[Dot[p, e]]]; Join[{1}, Select[Range[2, 500], fQ]] (* T. D. Noe, May 30 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), 3); \\ Michel Marcus, May 02 2018

Crossrefs

Cf. A000217, A001414, A212831.

Sequence in context: A264898 A248194 A295289 * A191487 A176205 A099256

Adjacent sequences: A212846 A212847 A212848 * A212850 A212851 A212852

Keyword

nonn,easy

Author

Jonathan Vos Post, May 28 2012

Status

approved