A110835 Smallest m > 0 such that there are no primes between n*m and n*(m+1) inclusive.

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

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 were 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. - Charles R Greathouse IV, Oct 09 2010

Links

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

A. Schinzel and W. Sierpinski, "Sur certaines hypothèses concernant 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
(Python)
from sympy import nextprime
def a(n):
    m = 1
    while nextprime(n*m-1) <= n*(m+1): m += 1
    return m
print([a(n) for n in range(1, 58)]) # Michael S. Branicky, Aug 04 2021

Crossrefs

See A014085 for primes between squares.

Sequence in context: A368250 A195344 A202998 * A087015 A200224 A345929

Adjacent sequences: A110832 A110833 A110834 * A110836 A110837 A110838

Keyword

nonn

Author

Franklin T. Adams-Watters, Sep 16 2005

Status

approved