[seqfan] Prime constellation asymptotics

Charles Greathouse charles.greathouse at case.edu
Tue Jul 24 10:03:21 CEST 2012

A008407 is a fundamental sequence in the study of prime
constellations: the minimum width of n primes in an interval allowed
by divisibility considerations. (That this corresponds to the minimum
width of a configuration of n primes is a deep problem in number
theory.)  What is known about the rate of its growth?

Clearly it must be superlinear.  Further, p# consecutive numbers
cannot contain more than (2-1)(3-1)(5-1)...(p-1) primes in any case,
so by Mertens' Theorem a(n) >> n log log n (with e^gamma as the
constant, of course).  But the sequence seems to grow faster, more
like n log n:

I can't prove any useful upper bounds on this sequence.

Charles Greathouse
Case Western Reserve University

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