A193990 -id:A193990 - cp's OEIS frontend

A108954 a(n) = pi(2*n) - pi(n). Number of primes in the interval (n,2n].

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

Offset: 1

Cino Hilliard, Jul 22 2005

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

F. Irschebeck, Einladung zur Zahlentheorie, BI Wissenschaftsverlag 1992, p. 40.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See pp. 181-182.

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, arXiv:0807.3690 [math.GM], 2008.

Cf. A000720, A060715.
Cf. A067434 (number of prime factors in binomial(2*n,n)), A193990, A074990.
Related sequences:
Primes (p) and composites (c): A000040, A002808, A000720, A065855.
Primes between p(n) and 2*p(n): A063124, A070046; between c(n) and 2*c(n): A376761; between n and 2*n: A035250, A060715, A077463, A108954.
Composites between p(n) and 2*p(n): A246514; between c(n) and 2*c(n): A376760; between n and 2*n: A075084, A307912, A307989, A376759.

A108954 := proc(n)
    numtheory[pi](2*n)-numtheory[pi](n) ;
end proc: # R. J. Mathar, Nov 03 2017

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 *)

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

from sympy import primepi
def A108954(n): return primepi(n<<1)-primepi(n) # Chai Wah Wu, Aug 19 2024

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
From Wesley Ivan Hurt, Sep 20 2021: (Start)
a(n) = Sum_{k=1..n} A010051(2*n-k+1).
a(n) = Sum_{k=n*(n+1)/2+2..(n+1)*(n+2)/2} A010051(A128076(k)). (End)

A067434 Number of distinct prime factors in binomial(2*n,n).

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 5, 5, 5, 6, 6, 6, 6, 7, 8, 8, 9, 9, 10, 10, 10, 9, 10, 10, 10, 10, 12, 13, 12, 12, 13, 14, 14, 14, 14, 14, 15, 14, 15, 15, 16, 16, 16, 17, 17, 17, 18, 18, 18, 18, 18, 19, 20, 19, 19, 19, 20, 20, 21, 21, 21, 21, 22, 22, 23, 24, 23, 23, 23, 23, 24, 24, 24, 25, 25

Offset: 1

Benoit Cloitre, Feb 23 2002

a(n) = A001221(A000984(n)) = length of n-th row in A226078. - Reinhard Zumkeller, May 25 2013

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A193990, A193991 (number of prime factors <= n and > n).

a067434 = a001221 . a000984 -- Reinhard Zumkeller, May 25 2013

a := n -> nops(numtheory:-factorset(binomial(2*n,n))):
seq(a(n), n=1..76); # Peter Luschny, Oct 31 2015

Table[Length[FactorInteger[Binomial[2 n, n]]], {n, 100}] (* T. D. Noe, Aug 17 2011 *)

a(n)=omega(binomial(2*n,n)) \\ Charles R Greathouse IV, May 25 2013

valp(n,p)=my(s);while(n\=p,s+=n);s
a(n)=my(s);forprime(p=2,2*n,if(valp(2*n,p)>2*valp(n,p),s++)); s \\ Charles R Greathouse IV, May 25 2013

from math import comb
from sympy import primenu
def A067434(n): return primenu(comb(n<<1,n)) # Chai Wah Wu, Aug 19 2024

a(n) ~ kn/log n, with k = log 4. - Charles R Greathouse IV, May 25 2013

cp's OEIS Frontend

A108954 a(n) = pi(2*n) - pi(n). Number of primes in the interval (n,2n].

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A067434 Number of distinct prime factors in binomial(2*n,n).

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Crossrefs

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Haskell

Maple

Mathematica

PARI

PARI

Python

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