A218850 a(n) is the least r > 1 for which the interval (r*(2*n-1), r*(2*n+1)) contains no prime, or 0 if no such r exists.

0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 5, 0, 2, 4, 2, 0, 4, 2, 3, 0, 0, 2, 3, 6, 0, 4, 0, 2, 2, 2, 0, 0, 3, 0, 2, 0, 7, 0, 2, 3, 16, 0, 2, 0, 2, 2, 3, 0, 3, 2, 2, 5, 2, 2, 8, 3, 0, 2, 0, 2, 2, 0, 7, 2, 4, 4, 0, 3, 0

Offset

1,10

Comments

In the first 50000 terms, the largest value is a(7333) = 37.

It is clear that a(1)=0, since it follows from the Bertrand postulate, which states that, for k>1, between k and 2*k there is a prime. This statement was proved first by P. Chebyshev and later by S. Ramanujan.

The equations a(2)=a(3)=a(4)=0 could be proved with the uniform positions, using Theorem 30 for generalized Ramanujan numbers from the Shevelev link. For proof of the equations a(n)=0 for n=5,...,9,11,13,17, etc., n<16597 we used a known result of L. Schoenfeld (1976) which states that for m>2010760, between m and m*(1+1/16597) there is always a prime, and, for 16597 <= n < 28314000, a stronger result of O. Ramaré and Y. Saouter (2003) which states that, for m >= 10726905041, between m*(1-1/28314000) and m there is always a prime.

Links

Peter J. C. Moses, Table of n, a(n) for n = 1..20000

J. Nagura, On the interval containing at least one prime number, Proc. Japan Acad., 28 (1952), 177-181.

S. Ramanujan, A proof of Bertrand's postulate, J. Indian Math. Soc., 11 (1919), 181-182.

O. Ramaré and Y. Saouter, Short effective intervals containing primes, J. Number Theory, 98(2003), 10-33.

L. Schoenfeld, Sharper bounds for the Chebyshev functions theta(x) and psi(x). II, Math. Comp. 30 (1975) 337-360.

Vladimir Shevelev, Ramanujan and Labos primes, their generalizations, and classifications of primes, J. Integer Seq. 15 (2012), Article 12.5.4.

Vladimir Shevelev, Charles R. Greathouse IV, and Peter J. C. Moses, On intervals (kn, (k+1)n) containing a prime for all n>1, Journal of Integer Sequences, Vol. 16 (2013), Article 13.7.3. arXiv:1212.2785 [math.NT], 2012.

Crossrefs

Cf. A218831, A218769.

Sequence in context: A204850 A202394 A202954 * A021627 A257580 A275933

Adjacent sequences: A218847 A218848 A218849 * A218851 A218852 A218853

Keyword

nonn

Author

Vladimir Shevelev and Peter J. C. Moses, Nov 07 2012

Status

approved