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On Intervals (***kn*, (*k* + 1)*n*) Containing a
Prime for All *n* > 1

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Vladimir Shevelev

Department of Mathematics

Ben-Gurion University of the Negev

Beer-Sheva 84105

Israel

Charles R. Greathouse IV

3214 Whitethorn Road

Cleveland Heights, OH 44118

USA

Peter J. C. Moses

Moparmatic Co.

1154 Evesham Road

Astwood Bank, Redditch, Worcestershire

B96 6DT England

**Abstract:**

We study values of *k* for which the interval
(*kn*, (*k* + 1)*n*) contains a prime for every *n* > 1.
We prove that the list of such integers *k*
includes 1, 2, 3, 5, 9, 14 and no others, at least for
*k* ≤ 100,000,000. Moreover, for every known *k* in this list,
we give a good upper bound for the
smallest *N*_{k}(*m*), such that if *n* >
*N*_{k}(*m*), then the interval
(*kn*, (*k* + 1)*n*)
contains at least *m* primes.

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(Concerned with sequences
A084140
A104272
A164952
A185004
A185005
A185006
A185007
A218769
A218831
A218850
A220268
A220269
A220273
A220274
A220281
A220293
A220462
A220463
A220474
A220475.)

Received January 1 2013;
revised version received May 7 2013; June 10 2013; July 25 2013.
Published in *Journal of Integer Sequences*, July 31 2013.

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