A073395 - OEIS

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

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OFFSET

1,2

LINKS

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

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.

EXAMPLE

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.

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 *)

PROG

(Haskell)

a073395 n = a008472 n * a001414 n

-- Reinhard Zumkeller, Oct 28 2015 (fixed), Mar 29 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

CROSSREFS