A023163 Numbers k such that Fibonacci(k) == -2 (mod k).

1, 9, 39, 111, 129, 159, 201, 249, 321, 471, 489, 519, 591, 681, 831, 849, 879, 921, 951, 1041, 1119, 1191, 1329, 1401, 1569, 1641, 1671, 1689, 1761, 1839, 1929, 1959, 2031, 2049, 2199, 2271, 2319, 2361, 2391, 2481, 2559, 2631, 2649, 2721, 2841, 2991, 3039

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Links

• Robert Israel, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A000045.

Programs

Maple

fpp:= n -> mpow(n-1,n)[2,2]:
M:= <<0,1>|<1,1>>:
mpow:= proc(n,p)
  if n = 0 then <<1,0>|<0,1>>
  elif n::even then procname(n/2,p)^2 mod p
  else  procname((n-1)/2,p)^2 . M mod p
  fi
end proc:
select(p -> fpp(p)+2 mod p = 0, [1, seq(i,i=3..10000,3)]); # Robert Israel, Feb 01 2017

Mathematica

Join[{1}, Position[Mod[Fibonacci[#], #]-#& /@ Range[10000], -2] // Flatten] (* Jean-François Alcover, Jun 09 2020 *)

PARI

isok(k) = Mod(fibonacci(k), k) == -2; \\ Michel Marcus, Jun 09 2020