A108046 Inverse Moebius transform of Fibonacci numbers 0, 1, 1, 2, 3, 5, 8, ...
0, 1, 1, 3, 3, 7, 8, 16, 22, 38, 55, 98, 144, 242, 381, 626, 987, 1625, 2584, 4221, 6774, 11002, 17711, 28768, 46371, 75170, 121415, 196662, 317811, 514650, 832040, 1346895, 2178365, 3525566, 5702898, 9229181, 14930352, 24160402, 39088314, 63250220, 102334155
Offset: 1
Keywords
Examples
a(4)=3 because the divisors of 4 are 1,2,4 and the first, second and fourth Fibonacci numbers are 0, 1 and 2, respectively, having sum 3.
Links
- Indranil Ghosh, Table of n, a(n) for n = 1..1000
Programs
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Maple
with(combinat): with(numtheory): f:=n->fibonacci(n-1): g:=proc(n) local div: div:=divisors(n): sum(f(div[j]),j=1..tau(n)) end: seq(g(n),n=1..45);
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Mathematica
a[n_] := DivisorSum[n, Fibonacci[#-1]&]; Array[a, 40] (* Jean-François Alcover, Dec 17 2015 *)
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PARI
a(n)=if(n<1,1,sumdiv(n,d,fibonacci(d-1))); /* Joerg Arndt, Aug 14 2012 */
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Python
from sympy import fibonacci, divisors def a(n): return 1 if n<1 else sum([fibonacci(d - 1) for d in divisors(n)]) # Indranil Ghosh, May 23 2017
Formula
G.f.: Sum_{k>=1} Fibonacci(k-1)*x^k/(1 - x^k). - Ilya Gutkovskiy, May 23 2017
a(n) = Sum_{d|n} Fibonacci(d-1). - Ridouane Oudra, Apr 11 2025