A081320 Largest 3-smooth divisor of n-th Fibonacci number.

1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 144, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 288, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 432, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 576, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 144, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 864, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 144, 1, 1, 2, 3, 1, 8, 1, 3, 2, 1, 1, 1152, 1

Offset

1,3

Comments

Conjecture: for n>12 and n>0 modulo 12: a(n)=a(n-12) and a(12*k)=A065331(k)*144.

The first part of the conjecture follows from the fact that the Fibonacci numbers are a strong divisibility sequence. - Charles R Greathouse IV, Sep 24 2012

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

a(n) = A065331(A000045(n)).

Example

Fibonacci(36) = 14930352 = 2^4 * 3^3 * 17 * 19 * 107, therefore a(36) = 2^4 * 3^3 = 432.

Mathematica

a[n_] := Times @@ ({2, 3}^IntegerExponent[Fibonacci[n], {2, 3}]);
Table[a[n], {n, 1, 1000}] (* Jean-François Alcover, Oct 15 2021 *)

Prog

(PARI) fibord(n, p)=if(n==0, return(oo)); my(u=3, t); while((t=((Mod([1, 1; 1, 0], p^u))^n)[1, 2])==0, u*=2); valuation(t, p)
a(n)=if(n%12, return(gcd(fibonacci(n%12), 24))); 3^fibord(n, 3)<<fibord(n, 2) \\ Charles R Greathouse IV, Nov 13 2015

Crossrefs

Cf. A003586, A065331, A000045.

Sequence in context: A097710 A171024 A109198 * A147865 A266614 A175314

Adjacent sequences: A081317 A081318 A081319 * A081321 A081322 A081323

Keyword

nonn

Author

Reinhard Zumkeller, May 20 2003

Status

approved