A069106 Composite numbers k such that k divides F(k-1) where F(j) are the Fibonacci numbers.

442, 1891, 2737, 4181, 6601, 6721, 8149, 13201, 13981, 15251, 17119, 17711, 30889, 34561, 40501, 51841, 52701, 64079, 64681, 67861, 68101, 68251, 78409, 88601, 88831, 90061, 96049, 97921, 115231, 118441, 138601, 145351, 146611, 150121, 153781, 163081, 179697, 186961, 191351, 194833

Offset: 1

Author

Benoit Cloitre, Apr 06 2002

Keywords

• easy
• nice
• nonn

Comments

Primes p congruent to 1 or 4 (mod 5) divide F(p-1) (cf. A045468 and [Hardy and Wright]).

References

• G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Fifth edition), Oxford Univ. Press (Clarendon), 1979, Chap. X, p. 150.

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1000

Crossrefs

Subsequence of A123976.

Cf. A045468, A003631, A064739, A081264 (Fibonacci pseudoprimes).

Cf. A002808, A000045.

Programs

C

#include  #include  #define STARTN 10 #define N_OF_MILLER_RABIN_TESTS 5 int main() { mpz_t n, f1, f2; int flag=0; /* flag? 0: f1 contains current F[n-1] 1: f2 = F[n-1] */ mpz_set_ui (n, STARTN); mpz_init (f1); mpz_init (f2); mpz_fib2_ui (f1, f2, STARTN-1); for (;;) { if (mpz_probab_prime_p (n, N_OF_MILLER_RABIN_TESTS)) goto next_iter; if (mpz_divisible_p (!flag? f1:f2, n)) { mpz_out_str (stdout, 10, n); printf (" "); fflush (stdout); } next_iter: mpz_add_ui (n, n, 1); mpz_add (!flag? f2:f1, f1, f2); flag = !flag; } }

Haskell

a069106 n = a069106_list !! (n-1)
a069106_list = [x | x <- a002808_list, a000045 (x-1) `mod` x == 0]
-- Reinhard Zumkeller, Jul 19 2013

Mathematica

A069106[nn_] := Select[Complement[Range[2,nn],Prime[Range[2,PrimePi[ nn]]]],Divisible[ Fibonacci[ #-1],#]&] (* Harvey P. Dale, Jul 05 2011 *)

PARI

fibmod(n,m)=((Mod([1,1;1,0],m))^n)[1,2]
is(n)=!isprime(n) && !fibmod(n-1,n) && n>1 \\ Charles R Greathouse IV, Oct 06 2016

Extensions

Corrected and extended (with C program) by Ralf Stephan, Oct 13 2002

a(35)-a(40) added by Reinhard Zumkeller, Jul 19 2013