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A181894 Sum of factors from A050376 in Fermi-Dirac representation of n. 2
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified October 21 22:47 EDT 2019. Contains 328315 sequences. (Running on oeis4.)