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

1,1

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

Mathematica

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

Prog

(C) #include <stdio.h> #include <gmp.h> #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
(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

Crossrefs

Subsequence of A123976.

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

Cf. A002808, A000045.

Sequence in context: A332531 A158322 A031720 * A094410 A236706 A105922

Adjacent sequences: A069103 A069104 A069105 * A069107 A069108 A069109

Keyword

easy,nice,nonn

Author

Benoit Cloitre, Apr 06 2002

Extensions

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

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

Status

approved