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A182504 Composite numbers k that divide both Fibonacci(k+1) and Fibonacci(2k+1)-1.

%I A182504 #48 Aug 02 2024 19:37:01
%S A182504 323,377,3827,5777,6479,10877,11663,18407,19043,20999,23407,25877,
%T A182504 27323,34943,35207,39203,44099,47519,50183,51983,53663,60377,65471,
%U A182504 75077,78089,79547,80189,81719,82983,84279,84419,86063,90287,94667,100127,104663,109871
%N A182504 Composite numbers k that divide both Fibonacci(k+1) and Fibonacci(2k+1)-1.
%C A182504 A subset of A182554 based on a refinement of the Fibonacci criterion for primality described there. The additional constraint that k divides Fibonacci(2*k+1)-1 is suggested by the Cloitre comment in A003631.
%C A182504 What base-2 pseudoprimes are contained in this sequence?
%C A182504 An almost identical sequence can be obtained by testing for composite numbers for which (1) k divides Fibonacci(k+1) and (2) k^12 mod 210 = 1. All primes greater than 7 appear to satisfy condition 2. Terms of {a(n)} which are not pseudoprimes to this criterion are 50183, 65471, 82983, and 84279. - _Gary Detlefs_, Jun 04 2012
%H A182504 Giovanni Resta, <a href="/A182504/b182504.txt">Table of n, a(n) for n = 1..1000</a>
%p A182504 with (combinat): f:= n-> fibonacci(n): for n from 2 to 100000 do if not isprime(n) and irem(f(n+1), n)=0 and irem((f(2*n+1)-1), n)=0 then print(n) fi od;
%t A182504 Select[Range[110000],CompositeQ[#]&&Mod[Fibonacci[#+1],#]==Mod[Fibonacci[ 2#+1]-1,#] == 0&] (* _Harvey P. Dale_, Aug 02 2024 *)
%o A182504 (Magma) [n: n in [4..11*10^4] | not IsPrime(n) and IsDivisibleBy(Fibonacci(n+1),n) and IsDivisibleBy(Fibonacci(2*n+1)-1,n)]; // _Bruno Berselli_, May 04 2012
%Y A182504 Cf. A182554, A003631, A038872, A000040.
%K A182504 nonn
%O A182504 1,1
%A A182504 _Gary Detlefs_, May 05 2012