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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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

Original entry on oeis.org

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
Offset: 1

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Author

Amarnath Murthy, Oct 25 2002

Keywords

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

Examples

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

Links

Crossrefs

Cf. A000005, A063375, A000045, A105800.
Cf. A076985.
Cf. A068499.

Programs

  • 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)
  • 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 */

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

Extensions

Corrected and extended by Sascha Kurz, Jan 26 2003
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.

A076985 Smallest Fibonacci numbers having exactly n Fibonacci divisors.

Original entry on oeis.org

1, 2, 8, 610, 144, 1134903170, 46368, 14930352, 4807526976, 1952132532477489958194625524584538730388053593825001030592563956919572392152809678530, 1548008755920, 303124944601307480151388995590175408058857436768033423077509087810
Offset: 1

Views

Author

Amarnath Murthy, Oct 25 2002

Keywords

Comments

a(n) = A000045(A105802(n)). - Reinhard Zumkeller, Apr 20 2005

Examples

			a(5) = 144 and the Fibonacci divisors are 1, 2, 3, 8 and 144.

Links

Crossrefs

Cf. A076984.

Extensions

More terms from Sascha Kurz, Jan 26 2003
a(12) from Reinhard Zumkeller, Apr 20 2005
Typo in data corrected by Sean A. Irvine, Apr 24 2025

A356666 Smallest m such that the m-th Lucas number has exactly n divisors that are also Lucas numbers.

Original entry on oeis.org

1, 0, 3, 6, 15, 30, 45, 90, 105, 210, 405, 810, 315, 630, 3645, 2025, 945, 1890, 1575, 3150, 2835, 5670, 36450, 25025, 3465, 6930, 101250, 11025, 22050, 51030, 14175, 28350, 10395, 20790, 2952450, 175175, 17325, 34650, 1937102445, 625625, 31185, 62370, 127575, 255150
Offset: 1

Views

Author

Michel Marcus, Aug 22 2022

Keywords

Comments

Further terms <= 51030: a(28) = 11025, a(29) = 22050, a(30) = 51030, a(31) = 14175, a(32) = 28350, a(33) = 10395, a(34) = 20790, a(37) = 17325, a(38) = 34650, a(41) = 31185, a(49) = 45045. - Daniel Suteu, Aug 24 2022

Links

Crossrefs

Cf. A000032, A038547, A102460, A304092, A356123.
Cf. A105802 (similar for Fibonacci).

Programs

  • PARI
    L(n)=fibonacci(n+1)+fibonacci(n-1); \\ A000032
isld(n) = { my(u1=1, u2=3, old_u1); if(n<=2, sign(n), while(n>u2, old_u1=u1; u1=u2; u2=old_u1+u2); (u2==n)); }; \\ A102460
nbld(n) = sumdiv(n, d, isld(d)); \\ A304092
a(n) = my(k=0); while(nbld(L(k)) != n, k++); k;
  • PARI
    countLd(n) = my(c=0,x=2,y=1); while(x<=n, if(n%x==0, c++); [x,y]=[y,x+y]); c;
a(n) = if(n==1, return(1)); my(k=0,x=2,y=1); while(1, if(countLd(x) == n, return(k)); [x,y,k]=[y,x+y,k+1]); \\ Daniel Suteu, Aug 24 2022

Formula

A000032(a(n)) = A356123(n).

Extensions

a(12)-a(26) from Daniel Suteu, Aug 24 2022
More terms from Daniel Suteu and David A. Corneth, Sep 04 2022
Showing 1-3 of 3 results.

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