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

%I A073395 #26 Dec 21 2021 23:30:36
%S A073395 0,4,9,8,25,25,49,12,18,49,121,35,169,81,64,16,289,40,361,63,100,169,
%T A073395 529,45,50,225,27,99,841,100,961,20,196,361,144,50,1369,441,256,77,
%U A073395 1681,144,1849,195,88,625,2209,55,98,84,400,255,2809,55,256,117,484
%N A073395 Product of sums of prime factors of n: with and without repetition.
%H A073395 T. D. Noe, <a href="/A073395/b073395.txt">Table of n, a(n) for n = 1..10000</a>
%F A073395 a(n) = A008472(n)*A001414(n).
%F A073395 a(m) is a square for all squarefree numbers m.
%F A073395 a(p^k) = k*p^2 for primes p.
%e A073395 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.
%t A073395 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 *)
%o A073395 (Haskell)
%o A073395 a073395 n = a008472 n * a001414 n
%o A073395 -- _Reinhard Zumkeller_, Oct 28 2015 (fixed), Mar 29 2012
%o A073395 (Python)
%o A073395 from sympy import primefactors, factorint
%o A073395 def a001414(n):
%o A073395     f=factorint(n)
%o A073395     return sum([i*f[i] for i in f])
%o A073395 def a(n): return 0 if n==1 else sum(primefactors(n))*a001414(n)
%o A073395 print([a(n) for n in range(1, 101)]) # _Indranil Ghosh_, Jun 23 2017
%Y A073395 Cf. A073396.
%Y A073395 Cf. A027746, A027748.
%Y A073395 Cf. A008472, A001414.
%K A073395 nonn,nice
%O A073395 1,2
%A A073395 _Reinhard Zumkeller_, Jul 30 2002