cp's OEIS Frontend

A182554 Composite numbers k that divide Fibonacci(k+1) or Fibonacci(k-1).

%I A182554 #51 Sep 12 2022 03:46:27
%S A182554 323,377,442,1891,2737,2834,3827,4181,5777,6479,6601,6721,8149,10877,
%T A182554 11663,13201,13981,15251,17119,17711,18407,19043,20999,23407,25877,
%U A182554 27323,30889,34561,34943,35207,39203,40501,44099,47519,50183,51841,51983,52701,53663
%N A182554 Composite numbers k that divide Fibonacci(k+1) or Fibonacci(k-1).
%C A182554 Pseudoprimes to a Fibonacci criterion for primality.
%C A182554 It is known that for prime p <> 5, Fibonacci(p-1) or Fibonacci(p+1) is divisible by p. (see Burton reference)
%C A182554 Primes for which Fibonacci(p-1) are divisible by p are congruent to {0,1,4} mod 5 and are listed in A038872.
%C A182554 Primes for which Fibonacci(p+1) are divisible by p are congruent to {2,3} mod 5 and are listed in A003631.
%C A182554 For n <= 1000, a(n) is squarefree (see A005117). - _Dmitry Kamenetsky_, Jul 20 2015
%C A182554 Any nonsquarefree term is divisible by the square of a Fibonacci-Wieferich prime (i.e., a prime p such that Fibonacci(j) == 0 (mod p^2) for some j not divisible by p). No Fibonacci-Wieferich primes are known, and there are none < 2*10^14, although it is conjectured that there are infinitely many. - _Robert Israel_, Jul 22 2015
%D A182554 David M. Burton, Elementary Number Theory, Allyn and Bacon, 1980, p. 292, #1.
%H A182554 Amiram Eldar, <a href="/A182554/b182554.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from Giovanni Resta)
%H A182554 R. J. McIntosh and E. L. Roettger, <a href="http://dx.doi.org/10.1090/S0025-5718-07-01955-2">A search for Fibonacci-Wieferich and Wolstenholme primes</a>, Math. Comp. 76 (2007), 2087-2094.
%p A182554 with(combinat): f:= n-> fibonacci(n): for n from 1 to 40000 do if (f(n+1)/n = floor(f(n+1)/n) or f(n-1)/n=floor(f(n-1)/n)) and not isprime(n) then print(n) fi od;
%t A182554 Select[Range[50000], CompositeQ[#] && (Divisible[Fibonacci[# - 1], #] || Divisible[Fibonacci[# + 1], #]) &] (* _Amiram Eldar_, Sep 12 2022 *)
%o A182554 (PARI) p=2;forprime(q=3,1e5,for(n=p+1,q-1,t=Mod([1,1;1,0],n)^(n-1);if(t[1,2]==0 || (t*[1,1;1,0]^2)[1,2]==0,print1(n", ")));p=q) \\ _Charles R Greathouse IV_, May 05 2012
%Y A182554 Cf. A038872, A003631, A000040, A094395, A005117.
%K A182554 nonn
%O A182554 1,1
%A A182554 _Gary Detlefs_, May 04 2012