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Search: id:A076984
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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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+0
4
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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
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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 (noe(AT)sspectra.com), Jan 18 2006
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LINKS
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Eric Weisstein's World of Mathematics, Fibonacci Number
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FORMULA
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a(n)=A023645(n)+1 - T. D. Noe (noe(AT)sspectra.com), 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 (alex(AT)kolmogorov.com), 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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PROGRAM
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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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Cf. A000005, A063375, A000045, A105800.
Cf. A076985.
Cf. A068499.
Sequence in context: A106696 A131839 A143299 this_sequence A079085 A076869 A104307
Adjacent sequences: A076981 A076982 A076983 this_sequence A076985 A076986 A076987
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KEYWORD
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nonn
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AUTHOR
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Amarnath Murthy (amarnath_murthy(AT)yahoo.com), Oct 25 2002
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EXTENSIONS
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Corrected and extended by Sascha Kurz (sascha.kurz(AT)uni-bayreuth.de), Jan 26 2003
Edited by N. J. A. Sloane (njas(AT)research.att.com), Sep 14 2006. Some of the comments and formulae may need to be adjusted to reflect the new offset.
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