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A166968 Minimum k such that for all m >= k there is a prime p with m < p < m * (n+1)/n. 2
2, 8, 9, 24, 25, 32, 33, 48, 115, 116, 117, 118, 118, 140, 140, 141, 200, 212, 212, 213, 294, 294, 318, 318, 319, 319, 320, 320, 320, 524, 525, 525, 526, 526, 526, 527, 527, 528, 528, 1328, 1329, 1330, 1331, 1331, 1332, 1333, 1333, 1334, 1334, 1335, 1335 (list; graph; refs; listen; history; text; internal format)
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.

Sondow, Jonathan and Weisstein, Eric W., 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

Cf. A060715, A104272.

Sequence in context: A046681 A259672 A163619 * A075644 A088825 A337706

Adjacent sequences:  A166965 A166966 A166967 * A166969 A166970 A166971

KEYWORD

nonn

AUTHOR

Michael B. Porter, Oct 25 2009

EXTENSIONS

Edited by Eric M. Schmidt, Oct 21 2017

STATUS

approved

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Last modified June 20 02:28 EDT 2021. Contains 345154 sequences. (Running on oeis4.)