A181894 Sum of factors from A050376 in Fermi-Dirac representation of n.

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, 29, 10, 31, 18, 14, 19, 12, 13, 37, 21, 16, 11, 41, 12, 43, 15, 14, 25, 47, 19, 49, 27, 20, 17, 53, 14, 16, 13, 22, 31, 59, 12, 61, 33, 16, 20, 18, 16, 67, 21, 26

Offset

1,2

Comments

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).

For n > 1: a(n) = sum (A213925(n,k): k=1..A064547(n)). - Reinhard Zumkeller, Mar 20 2013

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Formula

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).

Example

For n=54, the Fermi-Dirac representation is 54=2*3*9, then a(54)=2+3+9=14.

Mathematica

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 *)

Prog

(Haskell)
a181894 1 = 0
a181894 n = sum $ a213925_row n  -- Reinhard Zumkeller, Mar 20 2013

Crossrefs

Cf. A050376, A001414, A008472.

Sequence in context: A134889 A303702 A319057 * A265535 A094802 A075084

Adjacent sequences: A181891 A181892 A181893 * A181895 A181896 A181897

Keyword

nonn

Author

Vladimir Shevelev, Mar 31 2012

Status

approved