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

%I A081320 #17 Oct 15 2021 12:43:26
%S A081320 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,
%T A081320 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,
%U A081320 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
%N A081320 Largest 3-smooth divisor of n-th Fibonacci number.
%C A081320 Conjecture: for n>12 and n>0 modulo 12: a(n)=a(n-12) and a(12*k)=A065331(k)*144.
%C A081320 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
%H A081320 Charles R Greathouse IV, <a href="/A081320/b081320.txt">Table of n, a(n) for n = 1..10000</a>
%F A081320 a(n) = A065331(A000045(n)).
%e A081320 Fibonacci(36) = 14930352 = 2^4 * 3^3 * 17 * 19 * 107, therefore a(36) = 2^4 * 3^3 = 432.
%t A081320 a[n_] := Times @@ ({2, 3}^IntegerExponent[Fibonacci[n], {2, 3}]);
%t A081320 Table[a[n], {n, 1, 1000}] (* _Jean-François Alcover_, Oct 15 2021 *)
%o A081320 (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)
%o A081320 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
%Y A081320 Cf. A003586, A065331, A000045.
%K A081320 nonn
%O A081320 1,3
%A A081320 _Reinhard Zumkeller_, May 20 2003