# primefree sequence

Consider the sequence defined by $a_{1}=20615674205555510$, $a_{2}=3794765361567513$ and $a_{n}=a_{n-1}+a_{n-2}$ for all $n>2$. As it has been verified not to contain any primes, it is called a . The initial terms must be coprime, or else the lack of primes is a trivial consequence of the initial terms sharing a divisor other than 1.

Any Fibonacci-like sequence will naturally exhibit some patterns in the factorizations of its terms in relation 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, $2|a_{3x+1}$, $3|a_{4x+2}$, $5|a_{5x+1}$, $7|a_{8x}$, etc. for a finite number of potential prime factors (and $x\geq 0$ 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 $a_{138}$ and a few others afterwards to be prime.

The example sequence is listed in A083216 of the OEIS.

## References

• 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 numbers. Electronic J. of Combinatorics 6, 1999.
• 3 H. S. Wilf. Letters to the Editor. Math. Mag. 63, 284, 1990.
Title primefree sequence PrimefreeSequence 2013-03-22 15:54:49 2013-03-22 15:54:49 CompositeFan (12809) CompositeFan (12809) 7 CompositeFan (12809) Definition msc 11B39