cp's OEIS Frontend

A005086 Number of Fibonacci numbers 1,2,3,5,... dividing n.

%I A005086 #50 Jul 02 2025 16:01:54
%S A005086 1,2,2,2,2,3,1,3,2,3,1,3,2,2,3,3,1,3,1,3,3,2,1,4,2,3,2,2,1,4,1,3,2,3,
%T A005086 2,3,1,2,3,4,1,4,1,2,3,2,1,4,1,3,2,3,1,3,3,3,2,2,1,4,1,2,3,3,3,3,1,3,
%U A005086 2,3,1,4,1,2,3,2,1,4,1,4,2,2,1,4,2,2,2,3,2,4,2,2,2,2,2,4,1,2,2,3,1,4,1,4,4
%N A005086 Number of Fibonacci numbers 1,2,3,5,... dividing n.
%C A005086 Indices of records are in A129655. - _R. J. Mathar_, Nov 02 2007
%H A005086 Charles R Greathouse IV, <a href="/A005086/b005086.txt">Table of n, a(n) for n = 1..10000</a>
%F A005086 a(n) <= A072649(n). - _Robert G. Wilson v_, Dec 10 2006
%F A005086 Equals A051731 * A010056. - _Gary W. Adamson_, Nov 06 2007
%F A005086 G.f.: Sum_{n>=2} x^F(n)/(1-x^F(n)) where F(n) = A000045(n). - _Joerg Arndt_, Jan 06 2015
%F A005086 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A079586 - 1 = 2.359885... . - _Amiram Eldar_, Dec 31 2023
%p A005086 with(combinat): for n from 1 to 200 do printf(`%d,`,sum(floor(n/fibonacci(k))-floor((n-1)/fibonacci(k)), k=2..15)) od:
%t A005086 f[n_] := Block[{k = 1}, While[Fibonacci[k] <= n, k++ ]; Count[ Mod[n, Array[ Fibonacci, k - 1]], 0] - 1]; Array[f, 105] (* _Robert G. Wilson v_, Dec 10 2006 *)
%o A005086 (PARI) isfib(n)=my(k=n^2); k+=(k+1)<<2; issquare(k) || (n>0 && issquare(k-8))
%o A005086 a(n)=sumdiv(n,d,isfib(d)) \\ _Charles R Greathouse IV_, Nov 06 2014
%o A005086 (Python)
%o A005086 from sympy import divisors
%o A005086 from sympy.ntheory.primetest import is_square
%o A005086 def A005086(n): return sum(1 for d in divisors(n,generator=True) if is_square(m:=5*d**2-4) or is_square(m+8)) # _Chai Wah Wu_, Mar 30 2023
%o A005086 (Python)
%o A005086 from itertools import count, takewhile
%o A005086 def F(f=1, g=1):
%o A005086     while True:
%o A005086         f, g = g, f+g;
%o A005086         yield f
%o A005086 def a(n):
%o A005086     return sum(1 for f in takewhile(lambda x: x<=n, F()) if n%f == 0)
%o A005086 print([a(n) for n in range(1, 106)]) # _Michael S. Branicky_, Apr 03 2023
%Y A005086 Cf. A000045, A038663, A051731, A010056, A072649, A079586, A129655.
%K A005086 nonn
%O A005086 1,2
%A A005086 _N. J. A. Sloane_
%E A005086 More terms from _James Sellers_, Feb 19 2001
%E A005086 Incorrect comment removed by _Charles R Greathouse IV_, Nov 06 2014