A086382 k divides F(k*n^2+1)-F(k+1) for 1<=k<=a(n) where F(k) is the k-th Fibonacci number.

2, 1, 2, 10, 1, 10, 2, 1, 2, 12, 1, 10, 2, 1, 2, 10, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 16, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 16, 1, 10, 2, 1, 2, 10, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 12, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 36, 1, 10, 2, 1, 2, 10, 1, 12, 2, 1, 2, 10, 1, 10, 2, 1, 2, 12, 1, 12

Offset

2,1

Comments

Record values: a(2) = 2, a(5) = 10, a(11) = 12, a(29) = 16, a(71) = 36, a(3079) = 58. The next record a(n), if any has n > 10^5. - Robert Israel, Oct 14 2024

Links

Robert Israel, Table of n, a(n) for n = 2..10000

Formula

a(3n)=1; a( A047235(n))=2

Maple

fibmod:= proc(k, m) uses LinearAlgebra:-Modular;
  local M;
  M:= Mod(m, <<0, 1>|<1, 1>>, integer[8]);
  MatrixPower(m, M, k)[1, 2]
end proc:
f:= proc(n) local k;
   for k from 2 do if fibmod(k*n^2+1, k) <> fibmod(k+1, k) then return k-1 fi od
end proc:
map(f, [$2..100]); # Robert Israel, Oct 14 2024

Prog

(PARI) a(n)=if(n<0, 0, m=1; while((fibonacci(m*n^2+1)-fibonacci(m+1))%m==0, m++); m-1)

Crossrefs

Sequence in context: A006126 A078357 A225432 * A249416 A062345 A077098

Adjacent sequences: A086379 A086380 A086381 * A086383 A086384 A086385

Keyword

nonn

Author

Benoit Cloitre, Sep 06 2003

Status

approved