A094401 Composite n such that n divides both Fibonacci(n-1) and Fibonacci(n) - 1.

2737, 4181, 6721, 13201, 15251, 34561, 51841, 64079, 64681, 67861, 68251, 90061, 96049, 97921, 118441, 146611, 163081, 179697, 186961, 194833, 197209, 219781, 252601, 254321, 257761, 268801, 272611, 283361, 302101, 303101, 327313, 330929

Offset: 1

Author

Eric Rowland, May 01 2004

Keywords

• nonn

Comments

Composite n such that Q^(n-1) = I (mod n), where Q is the Fibonacci matrix {{1,1},{1,0}} and I is the identity matrix. The identity is also true for the primes congruent to 1 or 4 (mod 5), which is sequence A045468. The period of Q^k (mod n) is the same as the period of the Fibonacci numbers F(k) (mod n), A001175. Hence the terms in this sequence are the composite n such that A001175(n) divides n-1. [T. D. Noe, Jan 09 2009]

Links

• Giovanni Resta, Table of n, a(n) for n = 1..1000 (first 200 terms from T. D. Noe)
• Eric Weisstein, MathWorld: Fibonacci Q-Matrix.

Crossrefs

Cf. A005845, A069106, A094394, A094400.

Programs

Mathematica

Select[Range[2, 50000], ! PrimeQ[ # ] && Mod[Fibonacci[ # - 1], # ] == 0 && Mod[Lucas[ # ] - 1, # ] == 0 &]

Extensions

More terms from Ryan Propper, Sep 24 2005