cp's OEIS Frontend

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.

Showing 1-3 of 3 results.

A065069 Numbers n such that Fibonacci(n) is not squarefree, but for all proper divisors k of n, Fibonacci(k) is squarefree.

Original entry on oeis.org

6, 25, 56, 91, 110, 153, 406, 703, 752, 820, 915, 979, 1431, 1892, 2147, 2701, 2943, 3029, 3422, 4378, 4556, 4753, 4970, 5513, 6394, 7868, 8841, 9453, 10712, 12403, 13508, 13546, 15051, 16256, 17030, 17267, 18023, 18721, 19503, 20827, 21206
Offset: 1

Views

Author

Len Smiley, Nov 07 2001

Keywords

Comments

These are first primitive indices m for which Fib(m) is squareful. Note that Fib(km) is divisible by Fib(m).
This sequence is closely related to A001602(n), which gives the index of the smallest Fibonacci number divisible by prime(n). It can be shown that the index of the first Fibonacci number divisible by prime(n)^2 is A001602(n)*prime(n). This sequence is the collection of numbers A001602(n)*prime(n) with multiples removed. For example, A001602(2)*prime(2) = 12, but all multiples of 12 will generated by 6, the first number in this sequence. The Mathematica code assumes that Fibonacci numbers do not have any square primitive prime factors -- an assumption whose truth is an open question. - T. D. Noe, Jul 24 2003
These are the primitive elements of A037917. - Charles R Greathouse IV, Feb 02 2014
Terms after a(12) are conjectures until the factorizations of F(1271), F(1273), etc. are completed. - Charles R Greathouse IV, Feb 02 2014
Three more factorizations are needed to get the next term: F(1423), F(1427), and F(1429). If these are each squarefree, a(13) = 1431. - Charles R Greathouse IV, Dec 09 2022

Examples

			a(1) = 6 because 2^2 divides Fibonacci(6) but no smaller Fibonacci number.

Links

Crossrefs

Cf. A037917 (all indices <= 240 for which Fib(m) is squareful).
Cf. A065106, A001602, A013929 (not squarefree).

Programs

  • Mathematica
    << NumberTheory`NumberTheoryFunctions`; a = {}; l = 0; Do[m = n; If[k = 1; While[k < l + 1 && !IntegerQ[ n/ a[[k]]], k++ ]; k > l, If[ !SquareFreeQ[ Fibonacci[n]], a = Append[a, n]; l++; Print[n]]], {n, 1, 480} ]
nLimit=50000; i=3; pMax=1; iMax=1; While[p=Transpose[FactorInteger[Fibonacci[i]]][[1, -1]]; i*ppMax, pMax=p; iMax=i]; i++ ]; nMax=PrimePi[pMax]; fs={}; Do[p=Prime[n]; k=1; found=False; While[found=(Mod[Fibonacci[k], p]==0); !found&&k*p0, j=i+1; While[j<=Length[fs], If[Mod[fs[[j]], n]==0, fs[[j]]=0]; j++ ]]; i++ ]; Select[fs, #>0&&#
  • PARI
    is_A065069(n)=!fordiv(n,k,k>1 && k1 \\

Extensions

One more term from Robert G. Wilson v, Nov 08 2001
More terms from T. D. Noe, Jul 24 2003

A065107 Start of the permutation of the primes in the order in which p^2 first appears as a factor of a number in the Fibonacci sequence.

Original entry on oeis.org

2, 3, 5, 7, 13, 11, 17, 19, 29, 23, 37, 47, 41, 61, 31, 89
Offset: 1

Views

Author

Len Smiley, Nov 21 2001

Keywords

Comments

Assuming that there are no square primitive factors in the Fibonacci sequence (an open question), then this sequence continues 53, 43, 113, 73, 109, 233, 59, 107, 199, 67, 97, 71, 101, 149, 79, 139, 83, 151, 281, 421, 211, 137, 103, 157, 307, 521. This is obtained by sorting the pairs (prime(n)*A001602(n), prime(n)) by the first position and noting the order of the primes in the second position. - T. D. Noe, Apr 15 2004

Crossrefs

Cf. A065069, A065106.
Cf. A001602 (smallest m such that prime(n) divides Fibonacci(m)).

A264008 Index of the smallest Fibonacci number divisible by prime(n)^2.

Original entry on oeis.org

6, 12, 25, 56, 110, 91, 153, 342, 552, 406, 930, 703, 820, 1892, 752, 1431, 3422, 915, 4556, 4970, 2701, 6162, 6972, 979, 4753, 5050, 10712, 3852, 2943, 2147, 16256, 17030, 9453, 6394, 5513, 7550, 12403, 26732, 28056, 15051, 31862, 16290, 36290, 18721, 19503, 4378, 8862, 49952, 51756, 26106, 3029, 56882, 28920
Offset: 1

Views

Author

R. J. Mathar, Oct 31 2015

Keywords

Links

Crossrefs

Cf. A065069, A065106.

Programs

  • Maple
    f:= proc(n) local p, phi,q,k,G,Fkm,Fk,M,W,m;
  p:= ithprime(n);
  if member(p mod 5, {1,4}) then
    phi:= rhs(op(msolve(x^2-x-1,p^2)[1]));
    q:= -1-phi mod p^2;
    return numtheory:-order(q,p^2);
  fi;
  G:= GF(p,2,alpha^2-alpha-1);
  q:= G:-ConvertIn(-1-alpha);
  k:= G:-order(q);
  Fkm:= combinat:-fibonacci(k-1) mod p^2;
  Fk:= combinat:-fibonacci(k) mod p^2;
  M:= <|>;
  W:= <0,1>;
  for m from 1 do
     W:= M . W mod p^2;
     if W[1] = 0 then return(m*k) fi
  od:
end proc:
f(3):= 25:
map(f, [$1..100]); # Robert Israel, Jan 04 2018
  • PARI
    a(n) = if(n==3, 25, my(p=prime(n)); fordiv(p^2-1, d, if(fibonacci(d)%p==0, return(d*p)))); \\ Altug Alkan, Oct 31 2015

Formula

a(n) = prime(n)*A001602(n).
a(n) = min{i: A001248(n) | A000045(i)}
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.