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Template:Sequence of the Day for July 17

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Intended for: July 17, 2012

Timetable

  • First draft entered by Alonso del Arte on April 29, 2011 based on a verbatim copy of a write-up from November 6, 2010. ✓
  • Draft reviewed by Daniel Forgues on April 30, 2011
  • Draft approved by June 14, 2012
Yesterday's SOTD * Tomorrow's SOTD

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A083216: Fibonacci-like primefree sequence.

{ 20615674205555510, 3794765361567513, 24410439567123023, 28205204928690536, ... }
Like the sequence of Fibonacci numbers, this sequence has two initial terms set, and the rest are given by the familiar recurrence relation
a (n) = a (n  − 1) + a (n  − 2), n   ≥   2
. The amazing thing about this sequence is that there are no prime numbers among any of its terms even though the initial terms
a (0) = 2 × 5 × 5623 × 366631232537
and
a (1) = 3 × 1264921787189171
are coprime.

And whereas the search for prime numbers seeks larger and larger primes, the search for primefree sequences seeks smaller initial terms. When in 1964, Ronald Graham (of Graham’s number fame) proved that this kind of sequence is possible, his example had initial terms with more than thirty digits each (in base 10). Donald Knuth found a 17-digit pair in 1990, and later that same year Herbert Wilf found this sequence with slightly smaller initial terms. The current record is a 12-digit pair found by John W. Nicol in 1999.