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

Author

Franklin T. Adams-Watters, Sep 16 2005

Keywords

• nonn

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

Examples

			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.
		

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

Crossrefs

See A014085 for primes between squares.

A228775 is an alternate version.

Programs

Mathematica

a[n_]:=Module[{m=1},Until[NextPrime[n*m-1]>n*(m+1),m++];m];Array[a,57] (* James C. McMahon, Aug 10 2025 *)

PARI

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

PARI

a(n)=for(m=if(n>101,12118,4),oo, if(nextprime(n*m)>n*(m+1), return(m))) \\ Charles R Greathouse IV, Mar 04 2025

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