Primefree sequence
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In mathematics, a primefree sequence is a sequence of integers that does not contain any prime numbers. More specifically, it generally means a Fibonacci-like sequence with composite but coprime initial terms that is infinite but contains no primes. To put it algebraically, the sequence defined by GCD(a1,a2) = 1, and for n > 2 the recurrence relation an = an − 2 + an − 1, is a sequence that contains no primes.
Perhaps the best known primefree sequence is the one found by Wilf, with initial terms
- a1 = 20615674205555510,a2 = 3794765361567513
(sequence A083216 in OEIS), which is presently the primefree sequence with the second smallest known initial terms.
The requirement that the initial terms be coprime is necessary for the question to be non-trivial. If we allow the initial terms to share a prime factor p (e.g., a1 = xp,a2 = yp), due to the distributive property of multiplication it is obvious that a3 = (x + y)p; indeed all terms of the sequence will be multiples of p, with the primefreeness being a trivial consequence of that.
The order of the initial terms is also important. In Paul Hoffman's biography of Paul Erdős, The man who loved only numbers, the Wilf sequence is cited but with the initial terms switched. The resulting sequence appears primefree for the first hundred terms or so, but term 138 is the 45-digit prime 439351292910452432574786963588089477522344721. A108156 gives other indexes of prime numbers in the Wilf-Hoffman sequence.
Two other primefree sequences are known: a1 = 331635635998274737472200656430763,a2 = 1510028911088401971189590305498785, found by Graham in 1964, and a1 = 62638280004239857,a2 = 49463435743205655 found by Knuth in 1990. The primefree sequence with smallest known initial terms is presently a1 = 407389224418,a2 = 76343678551, found by Nicol in 1999.
