Wilson’s primeth recurrence

Define px to be the x-th prime numberMathworldPlanetmath (for example, p15=47). Then define the recurrence a0=1 and an=pan-1 for n>0. This is Wilson’s primeth recurrence which results in the sequence 1, 2, 3, 5, 11, 31, 127, 709, 5381, 52711, 648391, 9737333, 174440041, … (A7097 in Sloane’s OEIS). Given the prime counting function π(x), the recurrence should check out thus: π(an)=an-1.

It suffices to mention Euclid’s proof that there are infinitely many primes to show that this recurrence is also infinite. However, the terms of this recurrence quickly become large enough to show the limitations of today’s computational devices. Robert G. Wilson provided Sloane with just 15 terms. The last of those was shown to be erroneous by Paul Zimmerman, who was able to extend the known sequence by just two more terms. In 2007, David Baugh discovered two more terms.


  • 1 N. J. A. Sloane, “My Favorite Integer SequencesSequences and their Applications (Proceedings of SETA ’98), Springer-Verlag, London, 1999, pp. 103-130.
Title Wilson’s primeth recurrence
Canonical name WilsonsPrimethRecurrence
Date of creation 2013-03-22 16:11:51
Last modified on 2013-03-22 16:11:51
Owner PrimeFan (13766)
Last modified by PrimeFan (13766)
Numerical id 6
Author PrimeFan (13766)
Entry type Definition
Classification msc 11A41
Synonym Wilson’s prime-th recurrence
Synonym primeth recurrence
Synonym prime-th recurrence
Synonym primeth sequence
Synonym prime-th sequence