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

6, 12, 25, 56, 110, 91, 153, 342, 552, 406, 930, 703, 820, 1892, 752, 1431, 3422, 915, 4556, 4970, 2701, 6162, 6972, 979, 4753, 5050, 10712, 3852, 2943, 2147, 16256, 17030, 9453, 6394, 5513, 7550, 12403, 26732, 28056, 15051, 31862, 16290, 36290, 18721, 19503, 4378, 8862, 49952, 51756, 26106, 3029, 56882, 28920

Offset: 1

Author

R. J. Mathar, Oct 31 2015

Keywords

• nonn
• look

Links

• Robert Israel, Table of n, a(n) for n = 1..3584

Crossrefs

Cf. A065069, A065106.

Programs

Maple

f:= proc(n) local p, phi,q,k,G,Fkm,Fk,M,W,m;
  p:= ithprime(n);
  if member(p mod 5, {1,4}) then
    phi:= rhs(op(msolve(x^2-x-1,p^2)[1]));
    q:= -1-phi mod p^2;
    return numtheory:-order(q,p^2);
  fi;
  G:= GF(p,2,alpha^2-alpha-1);
  q:= G:-ConvertIn(-1-alpha);
  k:= G:-order(q);
  Fkm:= combinat:-fibonacci(k-1) mod p^2;
  Fk:= combinat:-fibonacci(k) mod p^2;
  M:= <|>;
  W:= <0,1>;
  for m from 1 do
     W:= M . W mod p^2;
     if W[1] = 0 then return(m*k) fi
  od:
end proc:
f(3):= 25:
map(f, [$1..100]); # Robert Israel, Jan 04 2018

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

Formula

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

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