A214434 - OEIS

%I #23 Sep 12 2022 03:51:12

%S 6601,13981,30889,68101,219781,252601,332949,399001,512461,642001,

%T 721801,722261,741751,852841,873181,1024651,1141141,1193221,1207361,

%U 1533601,1690501,1735841,1857241,1909001,2085301,2100901,2165801,2603381,2704801,2757241,3186821,3568661

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

%C Pseudoprimes to a criterion for primality which tests that

%C 1. k divides Fibonacci(k+1) or Fibonacci(k-1) (see A182554, A081264), and

%C 2. 2^(k-1) == 1 (mod k) (see A001567).

%C All terms appear to be congruent to 1 or -1 (mod 5).

%C Terms that are not congruent to 1 or -1 (mod 5): 22711873, 40160737, 55462177, ... . - _Amiram Eldar_, Sep 12 2022

%H Amiram Eldar, <a href="/A214434/b214434.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..671 from Giovanni Resta)

%F Intersection of A182554 and A001567.

%e 6601 is in the sequence because the 6600th Fibonacci number is divisible by 6601 and 2^6600 = 1 mod 6601.

%p 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 Select[Range[1, 4*10^6, 2], CompositeQ[#] && PowerMod[2, # - 1, #] == 1 && (Divisible[Fibonacci[# - 1], #] || Divisible[Fibonacci[# + 1], #]) &] (* _Amiram Eldar_, Sep 12 2022 *)

%Y Cf. A182554, A081264, A001567.

%K nonn

%O 1,1

%A _Gary Detlefs_, Jul 17 2012