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A084139 a(n) is the largest number for which exactly n primes are bounded between a(n) and 2a(n) exclusively. 8
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 (list; graph; refs; listen; history; text; internal format)
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: A020757 A253195 A049693 * A092590 A065394 A124011

Adjacent sequences:  A084136 A084137 A084138 * A084140 A084141 A084142

KEYWORD

nonn

AUTHOR

Harry J. Smith, May 15 2003

STATUS

approved

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Last modified December 7 07:15 EST 2016. Contains 278841 sequences.