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A076984
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Number of Fibonacci numbers that are divisors of the n-th Fibonacci number.
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6
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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, 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, 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
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OFFSET
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1,3
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COMMENTS
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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
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LINKS
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FORMULA
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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
The number of divisors of n excluding 2.
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
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EXAMPLE
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n=12, A000045(12)=144: 5 of the 15 divisors of 144 are also Fibonacci numbers, a(12) = #{1, 2, 3, 8, 144} = 5.
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MAPLE
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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);
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MATHEMATICA
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Table[s=DivisorSigma[0, n]; If[OddQ[n], s, s-1], {n, 100}] (Noe)
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PROG
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(PARI) {a(n)=if(n<1, 0, numdiv(n)+n%2-1)} /* Michael Somos, Sep 03 2006 */
(PARI) {a(n)=if(n<1, 0, sumdiv(n, d, d!=2))} /* Michael Somos, Sep 03 2006 */
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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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.
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STATUS
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approved
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