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

1,1

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

Links

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

Formula

Intersection of A182554 and A001567.

Example

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

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 *)

Crossrefs

Cf. A182554, A081264, A001567.

Sequence in context: A186563 A252637 A164971 * A317247 A290281 A178213

Adjacent sequences: A214431 A214432 A214433 * A214435 A214436 A214437

Keyword

nonn

Author

Gary Detlefs, Jul 17 2012

Status

approved