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Index of the smallest Fibonacci number divisible by prime(n)^2.
1

%I #15 Jan 04 2018 21:04:06

%S 6,12,25,56,110,91,153,342,552,406,930,703,820,1892,752,1431,3422,915,

%T 4556,4970,2701,6162,6972,979,4753,5050,10712,3852,2943,2147,16256,

%U 17030,9453,6394,5513,7550,12403,26732,28056,15051,31862,16290,36290,18721,19503,4378,8862,49952,51756,26106,3029,56882,28920

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

%H Robert Israel, <a href="/A264008/b264008.txt">Table of n, a(n) for n = 1..3584</a>

%F a(n) = prime(n)*A001602(n).

%F a(n) = min{i: A001248(n) | A000045(i)}

%p f:= proc(n) local p, phi,q,k,G,Fkm,Fk,M,W,m;

%p p:= ithprime(n);

%p if member(p mod 5, {1,4}) then

%p phi:= rhs(op(msolve(x^2-x-1,p^2)[1]));

%p q:= -1-phi mod p^2;

%p return numtheory:-order(q,p^2);

%p fi;

%p G:= GF(p,2,alpha^2-alpha-1);

%p q:= G:-ConvertIn(-1-alpha);

%p k:= G:-order(q);

%p Fkm:= combinat:-fibonacci(k-1) mod p^2;

%p Fk:= combinat:-fibonacci(k) mod p^2;

%p M:= <<Fkm, Fk>|<Fk,(Fkm+Fk mod p^2)>>;

%p W:= <0,1>;

%p for m from 1 do

%p W:= M . W mod p^2;

%p if W[1] = 0 then return(m*k) fi

%p od:

%p end proc:

%p f(3):= 25:

%p map(f, [$1..100]); # _Robert Israel_, Jan 04 2018

%o (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 Cf. A065069, A065106.

%K nonn,look

%O 1,1

%A _R. J. Mathar_, Oct 31 2015