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

%I A264008 #15 Jan 04 2018 21:04:06
%S A264008 6,12,25,56,110,91,153,342,552,406,930,703,820,1892,752,1431,3422,915,
%T A264008 4556,4970,2701,6162,6972,979,4753,5050,10712,3852,2943,2147,16256,
%U A264008 17030,9453,6394,5513,7550,12403,26732,28056,15051,31862,16290,36290,18721,19503,4378,8862,49952,51756,26106,3029,56882,28920
%N A264008 Index of the smallest Fibonacci number divisible by prime(n)^2.
%H A264008 Robert Israel, <a href="/A264008/b264008.txt">Table of n, a(n) for n = 1..3584</a>
%F A264008 a(n) = prime(n)*A001602(n).
%F A264008 a(n) = min{i: A001248(n) | A000045(i)}
%p A264008 f:= proc(n) local p, phi,q,k,G,Fkm,Fk,M,W,m;
%p A264008   p:= ithprime(n);
%p A264008   if member(p mod 5, {1,4}) then
%p A264008     phi:= rhs(op(msolve(x^2-x-1,p^2)[1]));
%p A264008     q:= -1-phi mod p^2;
%p A264008     return numtheory:-order(q,p^2);
%p A264008   fi;
%p A264008   G:= GF(p,2,alpha^2-alpha-1);
%p A264008   q:= G:-ConvertIn(-1-alpha);
%p A264008   k:= G:-order(q);
%p A264008   Fkm:= combinat:-fibonacci(k-1) mod p^2;
%p A264008   Fk:= combinat:-fibonacci(k) mod p^2;
%p A264008   M:= <<Fkm, Fk>|<Fk,(Fkm+Fk mod p^2)>>;
%p A264008   W:= <0,1>;
%p A264008   for m from 1 do
%p A264008      W:= M . W mod p^2;
%p A264008      if W[1] = 0 then return(m*k) fi
%p A264008   od:
%p A264008 end proc:
%p A264008 f(3):= 25:
%p A264008 map(f, [$1..100]); # _Robert Israel_, Jan 04 2018
%o A264008 (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
%Y A264008 Cf. A065069, A065106.
%K A264008 nonn,look
%O A264008 1,1
%A A264008 _R. J. Mathar_, Oct 31 2015