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A073395 Product of sums of prime factors of n: with and without repetition. 2
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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: A253560 A050399 A072995 * A064549 A304203 A087687

Adjacent sequences:  A073392 A073393 A073394 * A073396 A073397 A073398

KEYWORD

nonn,nice

AUTHOR

Reinhard Zumkeller, Jul 30 2002

STATUS

approved

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Last modified June 21 04:47 EDT 2021. Contains 345355 sequences. (Running on oeis4.)