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

%I A092340 #18 Apr 13 2021 19:10:04
%S A092340 7,17,107,137,197,227,347,617,827,857,1277,1427,1487,1607,1667,1697,
%T A092340 1787,1877,1997,2027,2087,2237,2267,2657,2687,2707,3167,3257,3467,
%U A092340 3527,3557,3767,3917,4127,4157,4217,4337,4517,4547,4637,4787,4967,5417,5477
%N A092340 Prime numbers n such that n^2+2*n divides (Fibonacci(n^2) + Fibonacci(2*n)).
%C A092340 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
%C A092340 See link for proof of this. - _Robert Israel_, Apr 13 2021
%H A092340 Giovanni Resta, <a href="/A092340/b092340.txt">Table of n, a(n) for n = 1..10000</a> (first 182 terms from Robert G. Wilson v)
%H A092340 Robert Israel, <a href="/A092340/a092340.pdf">Proof that A181605 is a subsequence of A092340</a>
%p A092340 filter:= proc(n)
%p A092340   local M,A;
%p A092340   uses LinearAlgebra:-Modular;
%p A092340   if not isprime(n) then return false fi;
%p A092340   M:= Matrix(2,2,<<0,1>|<1,1>>,datatype=integer);
%p A092340   A:= MatrixPower(n,M,n^2) + MatrixPower(n,M,2*n);
%p A092340   if A[1,2] mod n <> 0 then return false fi;
%p A092340   A:= MatrixPower(n+2,M,n^2) + MatrixPower(n+2,M,2*n);
%p A092340   A[1,2] mod (n+2) = 0
%p A092340 end proc:
%p A092340 select(filter, [seq(i,i=3..10000,2)]); # _Robert Israel_, Apr 13 2021
%t A092340 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 *)
%o A092340 (PARI) forprime (i=1,2000,if(Mod(fibonacci(i^2)+fibonacci(2*i),i^2+2*i)==0,print1(i,",")))
%Y A092340 Cf. A000045.
%Y A092340 Cf. A181605. - _Robert G. Wilson v_, Nov 07 2010
%K A092340 nonn
%O A092340 1,1
%A A092340 Mohammed Bouayoun (bouyao(AT)wanadoo.fr), Mar 18 2004
%E A092340 Offset changed from 0 to 1 by _Robert G. Wilson v_, Nov 07 2010
%E A092340 More terms from _Robert G. Wilson v_, Nov 07 2010