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%I #20 Aug 09 2015 15:57:10
%S 0,2,3,4,5,5,7,6,9,7,11,7,13,9,8,16,17,11,19,9,10,13,23,9,25,15,12,11,
%T 29,10,31,18,14,19,12,13,37,21,16,11,41,12,43,15,14,25,47,19,49,27,20,
%U 17,53,14,16,13,22,31,59,12,61,33,16,20,18,16,67,21,26
%N Sum of factors from A050376 in Fermi-Dirac representation of n.
%C Fermi-Dirac analog of A008472. Also, since a(q)=q iff q is in A050376, then for n=prod{q is in A050376}q, we have a(n)=sum{q is in A050376}a(q). Therefore, it is natural to call a(n) the Fermi-Dirac integer logarithm of n (Cf. A001414).
%C For n > 1: a(n) = sum (A213925(n,k): k=1..A064547(n)). - _Reinhard Zumkeller_, Mar 20 2013
%H Reinhard Zumkeller, <a href="/A181894/b181894.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n)=A008472(n) iff n is squarefree; if n is squarefree, then also a(n)=A001414(n), but here conversely, generally speaking, is not true. For example, a(24)=A001414(24). More general, if n is duplicate or quadruplicate squarefree number, then also a(n)=A001414(n).
%e For n=54, the Fermi-Dirac representation is 54=2*3*9, then a(54)=2+3+9=14.
%t FermiDiracSum[n_] := Module[{e, ex, p, s}, If[n <= 1, 0, {p, e} = Transpose[FactorInteger[n]]; s = 0; Do[d = IntegerDigits[e[[i]], 2]; ex = DeleteCases[Reverse[2^Range[0, Length[d] - 1]] d, 0]; s = s + Total[p[[i]]^ex], {i, Length[e]}]; s]]; Table[FermiDiracSum[n], {n, 100}] (* _T. D. Noe_, Apr 05 2012 *)
%o (Haskell)
%o a181894 1 = 0
%o a181894 n = sum $ a213925_row n -- _Reinhard Zumkeller_, Mar 20 2013
%Y Cf. A050376, A001414, A008472.
%K nonn
%O 1,2
%A _Vladimir Shevelev_, Mar 31 2012
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