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

5777, 10877, 17261, 75077, 80189, 100127, 113573, 120581, 161027, 162133, 163059, 231703, 300847, 430127, 618449, 635627, 667589, 851927, 1033997, 1106327, 1256293, 1388903, 1697183, 1842581, 2263127, 2435423, 2512889, 2662277

Offset: 1

Author

Eric Rowland, May 01 2004

Keywords

• nonn

Comments

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

Are all terms squarefree? A counterexample can't be divisible by the square of a prime < 1000. - Robert Israel, Jul 17 2015

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

Links

• Giovanni Resta, Table of n, a(n) for n = 1..579 (terms < 4*10^9)
• R. Israel and N. Elkies, Fibonacci == -1 mod p^2, Mathematics StackExchange, July 2015.
• R. J. McIntosh and E. L. Roettger, A search for Fibonacci-Wieferich and Wolstenholme primes, Math. Comp. 76 (2007), 2087-2094.

Crossrefs

Cf. A005845, A094394.

Programs

Maple

with(combinat):test:=n->(fibonacci(n)+1) mod n= 0:
select(test and not isprime ,[seq(2*k+1,k=1..10000)]);
# Robert FERREOL, Jul 14 2015

Mathematica

Select[ Range[3, 300000, 2], !PrimeQ[ # ] && Mod[Fibonacci[ # ] + 1, # ] == 0 &]

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

PARI

is(n)=((Mod([1,1;1,0],n))^n)[1,2]==-1 && n%2 && !isprime(n) \\ Charles R Greathouse IV, Jul 20 2015

Extensions

a(6)-a(14) from Robert G. Wilson v, May 01 2004

More terms from Ryan Propper, Aug 03 2005