Wilson’s primeth recurrence

Define $p_{x}$ to be the $x$ -th prime number (for example, $p_{15}=47$ ). Then define the recurrence $a_{0}=1$ and $a_{n}=p_{a_{n-1}}$ 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 $\pi(x)$ , the recurrence should check out thus: $\pi(a_{n})=a_{n-1}$ .

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.

References

1 N. J. A. Sloane, “My Favorite Integer Sequences” Sequences 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