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

%I A110835 #37 Aug 10 2025 11:55:15
%S A110835 8,4,8,6,18,15,17,25,13,20,29,44,87,81,35,83,79,74,70,67,118,330,58,
%T A110835 223,172,229,179,471,292,360,506,367,586,577,645,545,424,743,503,637,
%U A110835 766,467,937,579,698,683,542,1443,641,628,616,604,2026,1661,571,1834,551
%N A110835 Smallest m > 0 such that there are no primes between n*m and n*(m+1) inclusive.
%C A110835 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 A110835 Sierpinski's conjecture (1958) is precisely that a(n) >= n for all n. - _Charles R Greathouse IV_, Oct 09 2010
%H A110835 Charles R Greathouse IV, <a href="/A110835/b110835.txt">Table of n, a(n) for n = 1..599</a>
%H A110835 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 A110835 Wikipedia, <a href="http://en.wikipedia.org/wiki/Oppermann%27s_conjecture">Oppermann's conjecture</a>
%e A110835 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.
%t A110835 a[n_]:=Module[{m=1},Until[NextPrime[n*m-1]>n*(m+1),m++];m];Array[a,57] (* _James C. McMahon_, Aug 10 2025 *)
%o A110835 (PARI) a(n)=local(m);m=1;while(nextprime(n*m)<=n*(m+1),m=m+1);m
%o A110835 (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
%o A110835 (Python)
%o A110835 from sympy import nextprime
%o A110835 def a(n):
%o A110835     m = 1
%o A110835     while nextprime(n*m-1) <= n*(m+1): m += 1
%o A110835     return m
%o A110835 print([a(n) for n in range(1, 58)]) # _Michael S. Branicky_, Aug 04 2021
%Y A110835 See A014085 for primes between squares.
%Y A110835 A228775 is an alternate version.
%K A110835 nonn
%O A110835 1,1
%A A110835 _Franklin T. Adams-Watters_, Sep 16 2005