This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A181894 #27 May 02 2025 03:20:11
%S A181894 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 A181894 29,10,31,18,14,19,12,13,37,21,16,11,41,12,43,15,14,25,47,19,49,27,20,
%U A181894 17,53,14,16,13,22,31,59,12,61,33,16,20,18,16,67,21,26
%N A181894 Sum of factors from A050376 in Fermi-Dirac representation of n.
%C A181894 Fermi-Dirac analog of A008472. Also, since a(q) = q iff q is in A050376, then for n = Product_{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).
%H A181894 Reinhard Zumkeller, <a href="/A181894/b181894.txt">Table of n, a(n) for n = 1..10000</a>
%F A181894 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 generally, if n is duplicate or quadruplicate squarefree number, then also a(n) = A001414(n).
%F A181894 For n > 1: a(n) = Sum_{k=1..A064547(n)} A213925(n,k). - _Reinhard Zumkeller_, Mar 20 2013
%e A181894 For n = 54, the Fermi-Dirac representation is 54 = 2*3*9, then a(54) = 2+3+9 = 14.
%t A181894 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 A181894 (Haskell)
%o A181894 a181894 1 = 0
%o A181894 a181894 n = sum $ a213925_row n -- _Reinhard Zumkeller_, Mar 20 2013
%o A181894 (PARI) a(n) = if(n == 1, 0, my(f = factor(n), p = f[, 1], e = f[, 2], s = 0, b); for(i = 1, #p, b = binary(e[i]); for(j = 0, #b-1, if(b[#b-j], s += p[i]^(2^j)))); s); \\ _Amiram Eldar_, May 02 2025
%Y A181894 Cf. A001414, A008472, A050376, A064547, A213925.
%K A181894 nonn
%O A181894 1,2
%A A181894 _Vladimir Shevelev_, Mar 31 2012