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A214434 Composite numbers k such that k divides Fibonacci(k+1) or Fibonacci(k-1) and 2^(k-1) == 1 (mod k).

%I A214434 #23 Sep 12 2022 03:51:12
%S A214434 6601,13981,30889,68101,219781,252601,332949,399001,512461,642001,
%T A214434 721801,722261,741751,852841,873181,1024651,1141141,1193221,1207361,
%U A214434 1533601,1690501,1735841,1857241,1909001,2085301,2100901,2165801,2603381,2704801,2757241,3186821,3568661
%N A214434 Composite numbers k such that k divides Fibonacci(k+1) or Fibonacci(k-1) and 2^(k-1) == 1 (mod k).
%C A214434 Pseudoprimes to a criterion for primality which tests that
%C A214434 1. k divides Fibonacci(k+1) or Fibonacci(k-1) (see A182554, A081264), and
%C A214434 2. 2^(k-1) == 1 (mod k) (see A001567).
%C A214434 All terms appear to be congruent to 1 or -1 (mod 5).
%C A214434 Terms that are not congruent to 1 or -1 (mod 5): 22711873, 40160737, 55462177, ... . - _Amiram Eldar_, Sep 12 2022
%H A214434 Amiram Eldar, <a href="/A214434/b214434.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..671 from Giovanni Resta)
%F A214434 Intersection of A182554 and A001567.
%e A214434 6601 is in the sequence because the 6600th Fibonacci number is divisible by 6601 and 2^6600 = 1 mod 6601.
%p A214434 with(combinat):f:= n-> fibonacci(n): for n from 1 to 2000000 do if(f(n+1) mod n = 0 or f(n-1) mod n = 0) and 2^(n-1) mod n = 1 and not isprime(n) then print(n) fi od;
%t A214434 Select[Range[1, 4*10^6, 2], CompositeQ[#] && PowerMod[2, # - 1, #] == 1 && (Divisible[Fibonacci[# - 1], #] || Divisible[Fibonacci[# + 1], #]) &] (* _Amiram Eldar_, Sep 12 2022 *)
%Y A214434 Cf. A182554, A081264, A001567.
%K A214434 nonn
%O A214434 1,1
%A A214434 _Gary Detlefs_, Jul 17 2012