A073395 Product of sums of prime factors of n: with and without repetition.

0, 4, 9, 8, 25, 25, 49, 12, 18, 49, 121, 35, 169, 81, 64, 16, 289, 40, 361, 63, 100, 169, 529, 45, 50, 225, 27, 99, 841, 100, 961, 20, 196, 361, 144, 50, 1369, 441, 256, 77, 1681, 144, 1849, 195, 88, 625, 2209, 55, 98, 84, 400, 255, 2809, 55, 256, 117, 484

Offset: 1

Author

Reinhard Zumkeller, Jul 30 2002

Keywords

• nonn
• nice

Examples

			a(15) = (3+5)*(3+5) = 8*8 = 64 (n squarefree); a(16) = (2)*(2+2+2+2) = 2*8 = 16 (n prime-power); a(17) = (17)*(17) = 289 (n prime); a(18) = (2+3)*(2+3+3) = 5*8 = 40.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A073396.

Cf. A027746, A027748.

Cf. A008472, A001414.

Programs

Haskell

a073395 n = a008472 n * a001414 n
-- Reinhard Zumkeller, Oct 28 2015 (fixed), Mar 29 2012

Mathematica

a[n_] := With[{fi = FactorInteger[n]}, Plus @@ fi[[All, 1]]*Plus @@ Apply[Times, fi, 1]]; a[1]=0; Table[a[n], {n, 1, 57}] (* Jean-François Alcover, Jul 19 2012 *)

Python

from sympy import primefactors, factorint
def a001414(n):
    f=factorint(n)
    return sum([i*f[i] for i in f])
def a(n): return 0 if n==1 else sum(primefactors(n))*a001414(n)
print([a(n) for n in range(1, 101)]) # Indranil Ghosh, Jun 23 2017

Formula

a(n) = A008472(n)*A001414(n).

a(m) is a square for all squarefree numbers m.

a(p^k) = k*p^2 for primes p.