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
KEYWORD
nonn
AUTHOR
Franklin T. Adams-Watters, Sep 16 2005
STATUS
approved