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

Author

Vladimir Shevelev, Mar 31 2012

Keywords

• nonn

Comments

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).

Examples

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

Links

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

Crossrefs

Cf. A001414, A008472, A050376, A064547, A213925.

Programs

Haskell

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

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 *)

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

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 generally, if n is duplicate or quadruplicate squarefree number, then also a(n) = A001414(n).

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