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A108954 a(n) = pi(2*n) - pi(n). Number of primes in the interval (n,2n]. 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 (list; graph; refs; listen; history; text; internal format)
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

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>=i|i+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 A029170 A079526

Adjacent sequences:  A108951 A108952 A108953 * A108955 A108956 A108957

KEYWORD

nonn,easy

AUTHOR

Cino Hilliard, Jul 22 2005

STATUS

approved

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Last modified November 21 05:01 EST 2017. Contains 294988 sequences.