

A108954


a(n) = pi(2*n)  pi(n).


7



1, 1, 1, 2, 1, 2, 2, 2, 3, 4, 3, 4, 3, 3, 4, 5, 4, 4, 4, 4, 5, 6, 5, 6, 6, 6, 7, 7, 6, 7, 7, 7, 7, 8, 8, 9, 9, 9, 9, 10, 9, 10, 9, 9, 10, 10, 9, 9, 10, 10, 11, 12, 11, 12, 13, 13, 14, 14, 13, 13, 12, 12, 12, 13, 13, 14, 13, 13, 14, 15, 14, 14, 13, 13, 14, 15, 15, 15, 15, 15, 15, 16, 15, 16
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OFFSET

1,4


COMMENTS

a(n) < log(4)*n/log(n) < 7*n/(5*log(n)) for n > 1.  Reinhard Zumkeller, Mar 04 2008
Bertrand's postulate is equivalent to the formula a(n) >= 1 for all positive integers n.  Jonathan Vos Post, Jul 30 2008
Number of distinct prime factors > n of binomial(2*n,n).  T. D. Noe, Aug 18 2011
f(2, 2n)  f(3, n) < a(n) < f(3, 2n)  f(2, n) for n > 5889 where f(k, x) = x/log x * (1 + 1/log x + k/(log x)^2). The constant 3 can be improved.  Charles R Greathouse IV, May 02 2012


REFERENCES

F. Irschebeck, Einladung zur Zahlentheorie, BI Wissenschaftsverlag 1992, p. 40


LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000
Tsutomu Hashimoto, On a certain relation between Legendre's conjecture and Bertrand's postulate


FORMULA

Pi(x) = number of prime numbers less than or equal to x.
For n > 1, a(n) = A060715(n).  David Wasserman, Nov 04 2005
Conjecture: G.f.: Sum_{i>0} Sum_{j>=ii+j is prime} x^j.  Benedict W. J. Irwin, Mar 31 2017


MATHEMATICA

Table[Length[Select[Transpose[FactorInteger[Binomial[2 n, n]]][[1]], # > n &]], {n, 100}] (* T. D. Noe, Aug 18 2011 *)
f[n_] := Length@ Select[ Range[n + 1, 2n], PrimeQ]; Array[f, 100] (* Robert G. Wilson v, Mar 20 2012 *)
Table[PrimePi[2n]PrimePi[n], {n, 90}] (* Harvey P. Dale, Mar 11 2013 *)


PROG

(PARI) g(n) = for(x=1, n, y=primepi(2*x)primepi(x); print1(y", "))


CROSSREFS

a(n) = A000720(2*n)A000720(n).
Cf. A000720, A060715.
Cf. A067434 (number of prime factors in binomial(2*n,n)), A193990.
Sequence in context: A283190 A030361 A060715 * A123920 A029170 A079526
Adjacent sequences: A108951 A108952 A108953 * A108955 A108956 A108957


KEYWORD

nonn,easy


AUTHOR

Cino Hilliard, Jul 22 2005


STATUS

approved



