A076984 - OEIS

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A076984

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

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

1,3

COMMENTS

a(A001605(n)) = 2; a(A105802(n)) = n.

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

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Fibonacci Number

FORMULA

a(n) = A023645(n) + 1. - T. D. Noe, Jan 18 2006

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(2n) = A066660(n). a(2n-1) = A099774(n). - Michael Somos, Sep 03 2006

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

EXAMPLE

n=12, A000045(12)=144: 5 of the 15 divisors of 144 are also Fibonacci numbers, a(12) = #{1, 2, 3, 8, 144} = 5.

MAPLE

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);

MATHEMATICA

Table[s=DivisorSigma[0, n]; If[OddQ[n], s, s-1], {n, 100}] (Noe)

PROG

(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 */

CROSSREFS

Cf. A000005, A063375, A000045, A105800.

Cf. A076985.

Cf. A068499.

Sequence in context: A257497 A266225 A248596 * A079085 A076869 A335925

Adjacent sequences: A076981 A076982 A076983 * A076985 A076986 A076987

KEYWORD