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

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

1,1

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

Links

Table of n, a(n) for n=1..41.

Hisanori Mishima, Factorizations of Fibonacci numbers: n=1..100, n=101..200, n=201..300, n=301..400, n=401..480.

Blair Kelly, Fibonacci and Lucas Factorizations

Example

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

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*p<nLimit, If[p>pMax, 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*p<nLimit, k++ ]; If[found, AppendTo[fs, p*k]], {n, nMax}]; fs=Sort[fs]; i=1; While[i<=Length[fs]-1, n=fs[[i]]; If[n>0, j=i+1; While[j<=Length[fs], If[Mod[fs[[j]], n]==0, fs[[j]]=0]; j++ ]]; i++ ]; Select[fs, #>0&&#<nLimit&]

Prog

(PARI) is_A065069(n)=!fordiv(n, k, k>1 && k<n!=issquarefree(fibonacci(k)) && return) && n>1 \\

Crossrefs

Cf. A037917 (all indices <= 240 for which Fib(m) is squareful).

Cf. A065106, A001602, A013929 (not squarefree).

Sequence in context: A075224 A364928 A042185 * A319429 A022270 A001664

Adjacent sequences: A065066 A065067 A065068 * A065070 A065071 A065072

Keyword

nonn

Author

Len Smiley, Nov 07 2001

Extensions

One more term from Robert G. Wilson v, Nov 08 2001

More terms from T. D. Noe, Jul 24 2003

Status

approved