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A110835 Smallest m > 0 such that there are no primes between n*m and n*(m+1) inclusive. 6
8, 4, 8, 6, 18, 15, 17, 25, 13, 20, 29, 44, 87, 81, 35, 83, 79, 74, 70, 67, 118, 330, 58, 223, 172, 229, 179, 471, 292, 360, 506, 367, 586, 577, 645, 545, 424, 743, 503, 637, 766, 467, 937, 579, 698, 683, 542, 1443, 641, 628, 616, 604, 2026, 1661, 571, 1834, 551 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Suggested by Legendre's conjecture (still open) that there is always a prime between n^2 and (n+1)^2. If a(n)>=n+2, it implies that there is always a prime between n^2 and n*(n+1) and another between n*(n+1) and (n+1)^2. Note that the "inclusive" condition for the range affects only n=1. The value of a(1) would be 1 or 3 if this condition was taken to be exclusive or semi-inclusive, respectively. This is Oppermann's conjecture.

Sierpinski's conjecture (1958) is precisely that a(n) >= n for all n.

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..599

A. Schinzel and W. Sierpinski, "Sur certaines hypotheses concernment les nombres premiers", Acta Arithmetica 4 (1958), pp. 185-208.

Wikipedia, Oppermann's conjecture

EXAMPLE

a(2)=4 because the primes 3,5 and 7 are in range 2m to 2m+2 for m from 1 to 3, but 8, 9 and 10 are all composite.

PROG

(PARI) a(n)=local(m); m=1; while(nextprime(n*m)<=n*(m+1), m=m+1); m

CROSSREFS

See A014085 for primes between squares.

Sequence in context: A010524 A195344 A202998 * A087015 A200224 A124012

Adjacent sequences:  A110832 A110833 A110834 * A110836 A110837 A110838

KEYWORD

nonn

AUTHOR

Franklin T. Adams-Watters, Sep 16 2005

EXTENSIONS

Comment and reference to Sierpinski's (other) conjecture by Charles R Greathouse IV, Oct 09 2010

STATUS

approved

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Last modified July 4 12:18 EDT 2020. Contains 335448 sequences. (Running on oeis4.)