

A108954


a(n) = pi(2*n)  pi(n). Number of primes in the interval (n,2n].


12



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
For n >= 2, a(n) is the number of primes appearing in the 2nd row of an n X n square array whose elements are the numbers from 1..n^2, listed in increasing order by rows.  Wesley Ivan Hurt, May 17 2021


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

a(n) = A000720(2*n)A000720(n).
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


MAPLE

A108954 := proc(n)
numtheory[pi](2*n)numtheory[pi](n) ;
end proc: # R. J. Mathar, Nov 03 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

Cf. A000720, A060715.
Cf. A067434 (number of prime factors in binomial(2*n,n)), A193990, A074990.
Sequence in context: A283190 A030361 A060715 * A123920 A322141 A029170
Adjacent sequences: A108951 A108952 A108953 * A108955 A108956 A108957


KEYWORD

nonn,easy


AUTHOR

Cino Hilliard, Jul 22 2005


STATUS

approved



