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A094395 Odd composite n such that n divides Fibonacci(n) + 1.

%I A094395 #44 Jul 23 2015 08:38:49
%S A094395 5777,10877,17261,75077,80189,100127,113573,120581,161027,162133,
%T A094395 163059,231703,300847,430127,618449,635627,667589,851927,1033997,
%U A094395 1106327,1256293,1388903,1697183,1842581,2263127,2435423,2512889,2662277
%N A094395 Odd composite n such that n divides Fibonacci(n) + 1.
%C A094395 For each prime p, Fibonacci(p) = 5^((p-1)/2) mod p, so p divides Fibonacci(p) + 1 for each prime p=10k+-3. Hence it is interesting to seek also nonprimes with the same property, a motivation for this sequence. - _Robert FERREOL_, Jul 14 2015
%C A094395 Are all terms squarefree? A counterexample can't be divisible by the square of a prime < 1000. - _Robert Israel_, Jul 17 2015
%C A094395 Any term that is not squarefree must be divisible by the square of a Fibonacci-Wieferich prime (see the StackExchange link). There are believed to be infinitely many such primes, but none are known, and none are less than 2*10^14. - _Robert Israel_, Jul 22 2015
%H A094395 Giovanni Resta, <a href="/A094395/b094395.txt">Table of n, a(n) for n = 1..579</a> (terms < 4*10^9)
%H A094395 R. Israel and N. Elkies, <a href="http://math.stackexchange.com/questions/1365104/fibonacci-equiv-1-mod-p2">Fibonacci == -1 mod p^2</a>, Mathematics StackExchange, July 2015.
%H A094395 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 A094395 with(combinat):test:=n->(fibonacci(n)+1) mod n= 0:
%p A094395 select(test and not isprime ,[seq(2*k+1,k=1..10000)]);
%p A094395 # _Robert FERREOL_, Jul 14 2015
%t A094395 Select[ Range[3, 300000, 2], !PrimeQ[ # ] && Mod[Fibonacci[ # ] + 1, # ] == 0 &]
%o A094395 (PARI) main(size)=my(v=vector(size),i,t=1); for(i=1,size, while(1, if(t%2==1&&omega(t)>1&&(fibonacci(t)+1)%t==0, v[i]=t; break, t++)); t++); v; \\ _Anders Hellström_, Jul 17 2015
%o A094395 (PARI) is(n)=((Mod([1,1;1,0],n))^n)[1,2]==-1 && n%2 && !isprime(n) \\ _Charles R Greathouse IV_, Jul 20 2015
%Y A094395 Cf. A005845, A094394.
%K A094395 nonn
%O A094395 1,1
%A A094395 _Eric Rowland_, May 01 2004
%E A094395 a(6)-a(14) from _Robert G. Wilson v_, May 01 2004
%E A094395 More terms from _Ryan Propper_, Aug 03 2005