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

1,2

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

Cf. A073396.

Cf. A027746, A027748.

Cf. A008472, A001414.

Sequence in context: A050399 A072995 A354165 * A064549 A304203 A087687

Adjacent sequences: A073392 A073393 A073394 * A073396 A073397 A073398

Keyword

nonn,nice

Author

Reinhard Zumkeller, Jul 30 2002

Status

approved