A084139 a(n) is the largest number for which exactly n primes are bounded between a(n) and 2a(n) exclusively.

1, 5, 8, 14, 20, 23, 29, 33, 35, 48, 50, 53, 63, 74, 75, 83, 89, 90, 113, 114, 116, 119, 120, 131, 134, 140, 153, 155, 173, 174, 183, 186, 200, 204, 209, 215, 216, 219, 230, 243, 245, 251, 284, 285, 293, 296, 299, 300, 303, 320, 321, 323, 326, 329, 338, 359, 363

Offset: 0

Author

Harry J. Smith, May 15 2003

Keywords

• nonn

Comments

a(n) is the index of last occurrence of n in A060715. This calculation relies on the fact that Pi(2*m)-Pi(m) > m/(3*Log(m)) for m>=5. It can be shown that every integer >= 0 occurs in A060715, so there is no problem in finding the last occurrence.

A168421(n) = nextprime(a(n)), where nextprime(x) is the next prime > x. Note: some a(n) may be prime, therefore nextprime(x) not equal to x. - John W. Nicholson, Oct 11 2013

Examples

			a(10) = 50 since ten primes last arise between 50 and 100: 53, 59, 61, 67, 71, 73, 79, 83, 89, 97.
		

References

• P. Ribenboim, The Little Book of Big Primes. Springer-Verlag, 1991, p. 140.

Links

• T. D. Noe, Table of n, a(n) for n = 0..1000
• Eric W. Weisstein, MathWorld: Bertrand's Postulate

Crossrefs

Cf. A060715, A060756, A084138, A084140, A084141, A084142.

Programs

Mathematica

nn = 100; t = Table[0, {nn}]; Do[m = PrimePi[2*n] - PrimePi[n]; If[0 < m <= nn, t[[m]] = n], {n, 15*nn}]; Join[{1}, t] (* T. D. Noe, Dec 31 2012 *)

Formula

a(n) = floor((A104272(n)+1)/2) for n >= 1. - John W. Nicholson, Oct 11 2013

a(n) = A084140(n+1) - 1. - John W. Nicholson, Oct 11 2013