OFFSET
1,1
COMMENTS
The first term was proved by Chebyshev in 1850: for all m > 1, there is a prime number between m and 2m. It is known by Bertrand's Postulate after Joseph Bertrand, who first conjectured it in 1845, and also by Chebyshev's Theorem.
The result a(5)=25 was proved by Jitsuro Nagura in 1952.
The result a(16597)=2010760 was proved by Pierre Dusart in 1998.
LINKS
Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Pierre Dusart, Autour de la fonction qui compte le nombre de nombres premiers (French), 1998.
Jitsuro Nagura, On the interval containing at least one prime number, Proc. Japan Acad. 28: 177-181, 1952.
Eric Weisstein's World of Mathematics, Bertrand's Postulate.
Wikipedia, Bertrand's postulate.
EXAMPLE
For n=4, there are no primes between 23 and 23*5/4 = 28.75. But, for all m >= 24, there is a prime p such that m < p < 5m/4, so a(4) = 24.
PROG
(PARI) /* This function searches until it finds 10 primes between x and x*(n+1)/n */
pi_excl(y) = if(y==floor(y), primepi(y)-isprime(y), primepi(y)) /* all primes < y, primepi(y) is all primes <= y */
pbetween(x, y) = pi_excl(y) - primepi(x)
A166968(n) = {local(pr, x, r); pr=0; x=1; r=0; while(pr<10, pr=pbetween(x, x*(n+1)/n); if(pr==0, r=x+1); x=x+1); r}
(Sage)
def a_list() :
known_n, known_k = (16597, 2010760)
L = [0] * known_n
L[known_n-1] = known_k
for n in range(known_n-1, 0, -1) :
L[n-1] = 1 + next(k for k in range(L[n]-1, 0, -1) if next_prime(k) >= k*(n+1)/n)
return L
# Eric M. Schmidt, Oct 21 2017
CROSSREFS
KEYWORD
nonn
AUTHOR
Michael B. Porter, Oct 25 2009
EXTENSIONS
Edited by Eric M. Schmidt, Oct 21 2017
STATUS
approved