[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:
https://oeis.org/plot2a?name1=A008407&name2=A000027&tform1=untransformed&tform2=untransformed&shift=0&radiop1=ratio&drawpoints=true
I can't prove any useful upper bounds on this sequence.
Charles Greathouse
Analyst/Programmer
Case Western Reserve University
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