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

6601, 13981, 30889, 68101, 219781, 252601, 332949, 399001, 512461, 642001, 721801, 722261, 741751, 852841, 873181, 1024651, 1141141, 1193221, 1207361, 1533601, 1690501, 1735841, 1857241, 1909001, 2085301, 2100901, 2165801, 2603381, 2704801, 2757241, 3186821, 3568661

Offset: 1

Author

Gary Detlefs, Jul 17 2012

Keywords

• nonn

Comments

Pseudoprimes to a criterion for primality which tests that

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

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

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

Terms that are not congruent to 1 or -1 (mod 5): 22711873, 40160737, 55462177, ... . - Amiram Eldar, Sep 12 2022

Examples

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..671 from Giovanni Resta)

Crossrefs

Cf. A182554, A081264, A001567.

Programs

Maple

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;

Mathematica

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

Formula

Intersection of A182554 and A001567.