A110835 - OEIS

%I #28 Feb 03 2023 03:17:36

%S 8,4,8,6,18,15,17,25,13,20,29,44,87,81,35,83,79,74,70,67,118,330,58,

%T 223,172,229,179,471,292,360,506,367,586,577,645,545,424,743,503,637,

%U 766,467,937,579,698,683,542,1443,641,628,616,604,2026,1661,571,1834,551

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

%C 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.

%C Sierpinski's conjecture (1958) is precisely that a(n) >= n for all n. - _Charles R Greathouse IV_, Oct 09 2010

%H Charles R Greathouse IV, <a href="/A110835/b110835.txt">Table of n, a(n) for n = 1..599</a>

%H A. Schinzel and W. Sierpinski, "<a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa4/aa432.pdf">Sur certaines hypothèses concernant les nombres premiers</a>", Acta Arithmetica 4 (1958), pp. 185-208.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann%27s_conjecture">Oppermann's conjecture</a>

%e 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.

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

%o (Python)

%o from sympy import nextprime

%o def a(n):

%o     m = 1

%o     while nextprime(n*m-1) <= n*(m+1): m += 1

%o     return m

%o print([a(n) for n in range(1, 58)]) # _Michael S. Branicky_, Aug 04 2021

%Y See A014085 for primes between squares.

%K nonn

%O 1,1

%A _Franklin T. Adams-Watters_, Sep 16 2005