## You are here

Homeprimefree sequence

## Primary tabs

# 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 primefree sequence. 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.

## Mathematics Subject Classification

11B39*no label found*

- Forums
- Planetary Bugs
- HS/Secondary
- University/Tertiary
- Graduate/Advanced
- Industry/Practice
- Research Topics
- LaTeX help
- Math Comptetitions
- Math History
- Math Humor
- PlanetMath Comments
- PlanetMath System Updates and News
- PlanetMath help
- PlanetMath.ORG
- Strategic Communications Development
- The Math Pub
- Testing messages (ignore)

- Other useful stuff

## Recent Activity

new question: how to contest an entry? by zorba

new question: simple question by parag

Sep 26

new question: Latent variable by adam_reith

Sep 17

new question: Harshad Number by pspss

Sep 14

new problem: Geometry by parag