primefree sequence

Consider the sequenceMathworldPlanetmath defined by a1=20615674205555510, a2=3794765361567513 and an=an-1+an-2 for all n>2. As it has been verified not to contain any primes, it is called a primefree sequenceMathworldPlanetmath. The initial terms must be coprimeMathworldPlanetmath, or else the lack of primes is a trivial consequence of the initial terms sharing a divisorMathworldPlanetmathPlanetmath other than 1.

Any Fibonacci-like sequence will naturally exhibit some patterns in the factorizations of its terms in relationMathworldPlanetmath to their indices. The initial terms are chosen so that these patterns cover any possible value of n. So, for our example sequence, discovered by Wilf in 1990, 2|a3x+1, 3|a4x+2, 5|a5x+1, 7|a8x, etc. for a finite number of potential prime factorsMathworldPlanetmath (and x0 in each case).

Order is very important: switching the initial terms can cause primes to arise in the sequence. Switching the initial terms in our example causes a138 and a few others afterwards to be prime.

The example sequence is listed in A083216 of the OEIS.


  • 1 P. Hoffman. The Man Who Loved Only Numbers: The Story of Paul Erdos and the Search for Mathematical Truth. New York: Hyperion, 1998.
  • 2 H. Nicol. A Fibonacci-like sequence of composite numbersMathworldPlanetmath. Electronic J. of Combinatorics 6, 1999.
  • 3 H. S. Wilf. Letters to the Editor. Math. Mag. 63, 284, 1990.
Title primefree sequence
Canonical name PrimefreeSequence
Date of creation 2013-03-22 15:54:49
Last modified on 2013-03-22 15:54:49
Owner CompositeFan (12809)
Last modified by CompositeFan (12809)
Numerical id 7
Author CompositeFan (12809)
Entry type Definition
Classification msc 11B39