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

0,2

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

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

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

Example

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

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 *)

Crossrefs

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

Sequence in context: A314516 A314517 A314518 * A314519 A314520 A314521

Adjacent sequences: A084136 A084137 A084138 * A084140 A084141 A084142

Keyword

nonn

Author

Harry J. Smith, May 15 2003

Status

approved