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

Author

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

Keywords

• nonn

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.