Indices of Fibonacci numbers that do not have a factor of the form 4k+1.
0, 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48
1
The Fibonacci numbers can be extended to negative indices also. In that case, the negatives of these numbers can be added. It has been proved that there are no additional terms.
T. D. Noe, Plot of 12 terms
(Mma) Join[{0, 1, 2}, Select[Range[3, 50], Union[Mod[Transpose[FactorInteger[Fibonacci[#]]][[1]], 4]][[1]] > 1 &]]
Cf. A238329 (corresponding Fibonacci numbers).
nonn,fini,full
T. D. Noe, Jul 09 2014