

A212849


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


2



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
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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



