A092340 Prime numbers n such that n^2+2*n divides (Fibonacci(n^2) + Fibonacci(2*n)).

7, 17, 107, 137, 197, 227, 347, 617, 827, 857, 1277, 1427, 1487, 1607, 1667, 1697, 1787, 1877, 1997, 2027, 2087, 2237, 2267, 2657, 2687, 2707, 3167, 3257, 3467, 3527, 3557, 3767, 3917, 4127, 4157, 4217, 4337, 4517, 4547, 4637, 4787, 4967, 5417, 5477

Offset: 1

Author

Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 18 2004

Keywords

• nonn

Comments

First disagrees with A181605 at n=26: this sequence contains 2707, but A181605 doesn't. Is this a supersequence of A181605? - Nathaniel Johnston, Jun 25 2011

See link for proof of this. - Robert Israel, Apr 13 2021

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 182 terms from Robert G. Wilson v)
• Robert Israel, Proof that A181605 is a subsequence of A092340

Crossrefs

Cf. A000045.

Cf. A181605. - Robert G. Wilson v, Nov 07 2010

Programs

Maple

filter:= proc(n)
  local M,A;
  uses LinearAlgebra:-Modular;
  if not isprime(n) then return false fi;
  M:= Matrix(2,2,<<0,1>|<1,1>>,datatype=integer);
  A:= MatrixPower(n,M,n^2) + MatrixPower(n,M,2*n);
  if A[1,2] mod n <> 0 then return false fi;
  A:= MatrixPower(n+2,M,n^2) + MatrixPower(n+2,M,2*n);
  A[1,2] mod (n+2) = 0
end proc:
select(filter, [seq(i,i=3..10000,2)]); # Robert Israel, Apr 13 2021

Mathematica

fQ[n_] := Mod[ Fibonacci[n^2] + Fibonacci[2 n], n^2 + 2 n] == 0; Select[ Prime@ Range@ 744, fQ] (* Robert G. Wilson v, Nov 07 2010 *)

PARI

forprime (i=1,2000,if(Mod(fibonacci(i^2)+fibonacci(2*i),i^2+2*i)==0,print1(i,",")))

Extensions

Offset changed from 0 to 1 by Robert G. Wilson v, Nov 07 2010

More terms from Robert G. Wilson v, Nov 07 2010