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A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.

%I A076984 #25 Feb 16 2025 08:32:47
%S A076984 1,1,2,2,2,3,2,3,3,3,2,5,2,3,4,4,2,5,2,5,4,3,2,7,3,3,4,5,2,7,2,5,4,3,
%T A076984 4,8,2,3,4,7,2,7,2,5,6,3,2,9,3,5,4,5,2,7,4,7,4,3,2,11,2,3,6,6,4,7,2,5,
%U A076984 4,7,2,11,2,3,6,5,4,7,2,9,5,3,2,11,4,3,4,7,2,11,4,5,4,3,4,11,2,5,6,8,2,7,2
%N A076984 Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.
%C A076984 a(A001605(n)) = 2; a(A105802(n)) = n.
%C A076984 It is well known that if k is a divisor of n then F(k) divides F(n). Hence if n has d divisors, one expects that a(n)=d. However because F(1)=F(2)=1, there is one fewer Fibonacci divisor when n is even. So for even n, a(n)=d-1. - _T. D. Noe_, Jan 18 2006
%H A076984 T. D. Noe, <a href="/A076984/b076984.txt">Table of n, a(n) for n = 1..10000</a>
%H A076984 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FibonacciNumber.html">Fibonacci Number</a>
%F A076984 a(n) = A023645(n) + 1. - _T. D. Noe_, Jan 18 2006
%F A076984 a(n) = tau(n) - [n is even] = A000005(n) - A059841(n). Proof: gcd(Fib(m), Fib(n)) = Fib(gcd(m, n)) and Fib(2) = 1. - Olivier Wittenberg, following a conjecture of _Ralf Stephan_, Sep 28 2004
%F A076984 The number of divisors of n excluding 2.
%F A076984 a(2n) = A066660(n). a(2n-1) = A099774(n). - _Michael Somos_, Sep 03 2006
%F A076984 a(3*2^(Prime(n-1)-1)) = 2n + 1 for n > 3. a(3*2^A068499[n]) = 2n + 1, where A068499(n) = {1,2,3,4,6,10,12,16,18,...}. - _Alexander Adamchuk_, Sep 15 2006
%e A076984 n=12, A000045(12)=144: 5 of the 15 divisors of 144 are also Fibonacci numbers, a(12) = #{1, 2, 3, 8, 144} = 5.
%p A076984 with(combinat, fibonacci):a[1] := 1:for i from 2 to 229 do s := 0:for j from 2 to i do if((fibonacci(i) mod fibonacci(j))=0) then s := s+1:fi:od:a[i] := s:od:seq(a[l],l=2..229);
%t A076984 Table[s=DivisorSigma[0, n]; If[OddQ[n], s, s-1], {n, 100}] (Noe)
%o A076984 (PARI) {a(n)=if(n<1, 0, numdiv(n)+n%2-1)} /* _Michael Somos_, Sep 03 2006 */
%o A076984 (PARI) {a(n)=if(n<1, 0, sumdiv(n,d, d!=2))} /* _Michael Somos_, Sep 03 2006 */
%Y A076984 Cf. A000005, A063375, A000045, A105800.
%Y A076984 Cf. A076985.
%Y A076984 Cf. A068499.
%K A076984 nonn
%O A076984 1,3
%A A076984 _Amarnath Murthy_, Oct 25 2002
%E A076984 Corrected and extended by _Sascha Kurz_, Jan 26 2003
%E A076984 Edited by _N. J. A. Sloane_, Sep 14 2006. Some of the comments and formulas may need to be adjusted to reflect the new offset.