

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
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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. SpringerVerlag, 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: A249719 A020757 A049693 * A092590 A065394 A124011
Adjacent sequences: A084136 A084137 A084138 * A084140 A084141 A084142


KEYWORD

nonn


AUTHOR

Harry J. Smith, May 15 2003


STATUS

approved



