A060715 -id:A060715 - 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)

A056171 a(n) = pi(n) - pi(floor(n/2)), where pi is A000720.

Original entry on oeis.org

0, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 3, 2, 2, 2, 3, 3, 4, 4, 4, 3, 4, 4, 4, 3, 3, 3, 4, 4, 5, 5, 5, 4, 4, 4, 5, 4, 4, 4, 5, 5, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 7, 7, 7, 7, 7, 6, 7, 7, 8, 7, 7, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 9, 9, 9, 9, 9, 10, 10, 10, 9, 10, 10, 10, 9, 9, 9, 10, 10, 10, 10, 10, 9, 9, 9, 10, 10

Offset: 1

Labos Elemer, Jul 27 2000

Also, the number of unitary prime divisors of n!. A prime divisor of n is unitary iff its exponent is 1 in the prime power factorization of n. In general, gcd(p, n/p) = 1 or p. Here we count the cases when gcd(p, n/p) = 1.
A unitary prime divisor of n! is >= n/2, hence their number is pi(n) - pi(n/2). - Peter Luschny, Mar 13 2011
See also the references and links mentioned in A143227. - Jonathan Sondow, Aug 03 2008
From Robert G. Wilson v, Mar 20 2017: (Start)
First occurrence of k is at n = A080359(k).
The last occurrence of k is at n = A080360(k).
The number of times k appears is A080362(k). (End)
Lev Schnirelmann proved that for every n, a(n) > (1/log_2(n))*(n/3 - 4*sqrt(n)) - 1 - (3/2)*log_2(n). - Arkadiusz Wesolowski, Nov 03 2017

			10! = 2^8 * 3^2 * 5^2 * 7. The only unitary prime divisor is 7, so a(10) = 1.

Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 214.

Daniel Forgues, Table of n, a(n) for n=1..100000
Ethan Berkove and Michael Brilleslyper, Subgraphs of Coprime Graphs on Sets of Consecutive Integers, Integers, Vol. 22 (2022), #A47, see p. 8.

Cf. A000720, A001221, A034444, A048105, A048656, A048657, A056169, A056172.
Cf. A014085, A060715, A104272, A143223, A143224, A143225, A143226, A143227.
Cf. A080359, A080360, A080362.

A056171 := proc(x)
     numtheory[pi](x)-numtheory[pi](floor(x/2)) ;
end proc:
seq(A056171(n),n=1..130) ; # N. J. A. Sloane, Sep 01 2015
A056171 := n -> nops(select(isprime,[$iquo(n,2)+1..n])):
seq(A056171(i),i=1..98); # Peter Luschny, Mar 13 2011

s=0; Table[If[PrimeQ[k], s++]; If[PrimeQ[k/2], s--]; s, {k,100}]
Table[PrimePi[n]-PrimePi[Floor[n/2]],{n,100}] (* Harvey P. Dale, Sep 01 2015 *)

A056171=n->primepi(n)-primepi(n\2) \\ M. F. Hasler, Dec 31 2016

from sympy import primepi
[primepi(n) - primepi(n//2) for n in range(1,151)] # Indranil Ghosh, Mar 22 2017

[prime_pi(n)-prime_pi(floor(n/2)) for n in range(1,99)] # Stefano Spezia, Apr 22 2025

a(n) = A000720(n) - A056172(n). - Robert G. Wilson v, Apr 09 2017

a(n) = A056169(n!). - Amiram Eldar, Jul 24 2024

Definition simplified by N. J. A. Sloane, Sep 01 2015

A070046 Number of primes between prime(n) and 2*prime(n) exclusive.

Original entry on oeis.org

1, 1, 1, 2, 3, 3, 4, 4, 5, 6, 7, 9, 9, 9, 9, 11, 13, 12, 13, 14, 13, 15, 15, 16, 19, 20, 19, 19, 18, 18, 23, 23, 25, 25, 27, 26, 28, 28, 28, 28, 30, 30, 32, 32, 32, 32, 35, 38, 38, 38, 39, 39, 39, 41, 42, 43, 42, 42, 42, 42, 42, 44, 49, 50, 49, 49, 54, 54, 56, 55, 55, 55, 57, 58

Offset: 1

Enoch Haga, May 05 2002

			a(1)=1 because between p=2 and 4 there is exactly one prime, 3.
a(10)=6 since six consecutive primes (31,37,41,43,47,53) are located between p(10) = 29 and 58.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Bertrand's Postulate

Cf. A060715, A077463, A246514.
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.

N:= 1000: # to get a(n) for n <= pi(N)
Primes:=select(isprime,[$1..N]):
seq(numtheory:-pi(2*Primes[n])-n, n=1..nops(Primes)); # Robert Israel, Aug 28 2014

pp[n_]:=Module[{pr=Prime[n]},PrimePi[2pr]-n]; Array[pp,80] (* Harvey P. Dale, Mar 30 2015; edited by Zak Seidov, Oct 18 2022  *)

forprime(p=2, 5000, n=0; for(q=p+1, 2*p-1, if(isprime(q), n++)); print1(n, ", ")) \\ Harry J. Smith, Dec 13 2007, improved by Colin Barker, Aug 28 2014

a(n)=primepi(2*prime(n))-n \\ Charles R Greathouse IV, Aug 28 2014

from sympy import prime, primepi
def A070046(n): return primepi(prime(n)<<1)-n # Chai Wah Wu, Oct 22 2024

a(n) = primepi(2*prime(n))-n. - Charles R Greathouse IV, Aug 28 2014
a(n) = A060715(A000040(n)).
a(n) = A063124(n)-1. - N. J. A. Sloane, Oct 19 2024

Edited by N. J. A. Sloane, May 15 2008 at the suggestion of R. J. Mathar

A063124 a(n) = # { primes p | prime(n) <= p < 2*prime(n) } where prime(n) is the n-th prime.

Original entry on oeis.org

2, 2, 2, 3, 4, 4, 5, 5, 6, 7, 8, 10, 10, 10, 10, 12, 14, 13, 14, 15, 14, 16, 16, 17, 20, 21, 20, 20, 19, 19, 24, 24, 26, 26, 28, 27, 29, 29, 29, 29, 31, 31, 33, 33, 33, 33, 36, 39, 39, 39, 40, 40, 40, 42, 43, 44, 43, 43, 43, 43, 43, 45, 50, 51, 50, 50, 55, 55, 57, 56, 56, 56, 58

Offset: 1

Reinhard Zumkeller, Aug 08 2001

nonn

a(n) is the number of primes between prime(n) and 2*prime(n) inclusive. - Sean A. Irvine, Apr 18 2023

Also for x = Product_{i=n..n+k} A000040(i), the least k such that A003961(x) > 2*x. - Antti Karttunen, Dec 08 2024

			a(10) = 7 as there are 7 primes between prime(10) = 29 and 58 = 29*2: 29, 31, 37, 41, 43, 47, 53.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000 [First 2000 terms from Harry J. Smith]

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.
Cf. A003961, A108227, A331677, A334051, A378746.

A062134 := proc(n) numtheory:-pi(2*ithprime(n))-n+1; end; # N. J. A. Sloane, Oct 19 2024
[seq(A062134(n),n=1..100)];

Table[PrimePi[2*Prime[n]] - n + 1, {n, 100}] (* Paolo Xausa, Oct 22 2024 *)

a(n)={1 + primepi(2*prime(n)) - n} \\ Harry J. Smith, Aug 19 2009

a(n) = A035250(prime(n)).
a(n) = A070046(n) + 1. - Sean A. Irvine, Apr 18 2023
From Antti Karttunen, Dec 08 2024: (Start)
a(n) = n-A331677(n) = 1+n-A334051(n).
a(n) = 1+A000720(2*A000040(n))-n. [After Harry J. Smith's PARI-program]
a(n) < A108227(n). [Assuming M. F. Hasler's interpretation in May 08 2017 comment in the latter]
a(n) = A001222(A378746(n)).
(End)

Definition clarified by N. J. A. Sloane, Oct 04 2024

A075084 Number of composite numbers c with n <= c <= 2*n.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 5, 7, 7, 7, 8, 9, 10, 12, 12, 12, 13, 15, 15, 17, 17, 17, 18, 19, 20, 21, 21, 22, 23, 24, 24, 26, 27, 27, 28, 28, 28, 30, 31, 31, 32, 33, 34, 36, 36, 37, 38, 40, 40, 41, 41, 41, 42, 43, 43, 44, 44, 45, 46, 48, 49, 51, 52, 52, 53, 53, 54, 56, 56, 56, 57, 59, 60

Offset: 1

Amarnath Murthy, Sep 11 2002

The number of composite numbers <= n is n less the number of primes less 1.

The sequence is nondecreasing.

			a(8) = 7: the composite numbers are 8,9,10,12,14,15 and 16.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

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.

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855
A075084 := proc(n) chi(2*n) - chi(n-1); end;
a := [seq(A075084(n),n=1..120)]; # N. J. A. Sloane, Oct 20 2024

Table[n - PrimePi[2n] + PrimePi[n - 1] + 1, {n, 2, 75}]

a(n) = if (n>1, n - primepi(2*n) + primepi(n-1) + 1, 0); \\ Michel Marcus, Oct 21 2024

from sympy import primepi
def A075084(n): return n+primepi(n-1)-primepi(n<<1)+1 if n>1 else 0 # Chai Wah Wu, Oct 20 2024

a(n) = n - pi(2n) + pi(n-1) + 1, for n>1.

Edited by Robert G. Wilson v, Sep 12 2002

Definition clarified by N. J. A. Sloane, Oct 20 2024

A077463 Number of primes p such that n < p < 2n-2.

Original entry on oeis.org

0, 0, 0, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 3, 4, 4, 5, 5, 5, 6, 6, 6, 7, 6, 6, 6, 7, 7, 7, 8, 8, 8, 9, 9, 9, 9, 9, 9, 9, 9, 10, 9, 9, 9, 10, 10, 11, 11, 11, 12, 13, 13, 14, 13, 13, 12, 12, 12, 12, 13, 13, 13, 13, 13, 14, 14, 14, 13, 13, 13, 14, 15, 15, 14, 15, 15, 15, 15, 15

Offset: 1

Eric W. Weisstein, Nov 05 2002

nonn

a(n) > 0 for n > 3 by Bertrand's postulate (and Chebyshev's proof of 1852). - Jonathan Vos Post, Aug 08 2013

			a(19) = 3, the first value smaller than a previous value, because the only primes between 19 and 2 * 19 - 2 = 36 are {23,29,31}. - _Jonathan Vos Post_, Aug 08 2013

J. Sondow and E. Weisstein, Bertrand's Postulate, World of Mathematics
M. Tchebichef, Memoire sur les nombres premiers, J. Math. Pures Appliq. 17 (1852) 366.

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.

a[n_] := PrimePi[2n - 2] - PrimePi[n]; a[1] = 0; Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 31 2012 *)

A376759 Number of composite numbers c with n < c <= 2*n.

Original entry on oeis.org

0, 1, 2, 2, 4, 4, 5, 6, 6, 6, 8, 8, 10, 11, 11, 11, 13, 14, 15, 16, 16, 16, 18, 18, 19, 20, 20, 21, 23, 23, 24, 25, 26, 26, 27, 27, 28, 29, 30, 30, 32, 32, 34, 35, 35, 36, 38, 39, 39, 40, 40, 40, 42, 42, 42, 43, 43, 44, 46, 47, 49, 50, 51, 51, 52, 52, 54, 55, 55, 55, 57, 58, 60, 61, 61, 61, 62, 63, 64, 65, 66, 66, 68, 68, 69, 70, 70, 71, 73, 73, 73, 74, 75, 76, 77, 77

Offset: 1

N. J. A. Sloane, Oct 20 2024

nonn

This completes the set of four: A307912, A376759, A307989, and A075084. Since it is not clear which ones are the most important, and they are easily confused, all four are now in the OEIS.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000

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.

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855
A376759 := proc(n) chi(2*n) - chi(n); end;
a := [seq(A376759(n),n=1..120)];

Table[PrimePi[n] - PrimePi[2*n] + n, {n, 100}] (* Paolo Xausa, Oct 22 2024 *)

from sympy import primepi
def A376759(n): return n+primepi(n)-primepi(n<<1) # Chai Wah Wu, Oct 20 2024

a(n) = A000720(n) - A000720(2*n) + n. - Paolo Xausa, Oct 22 2024

A246514 Number of composite numbers between prime(n) and 2*prime(n) exclusive.

Original entry on oeis.org

0, 1, 3, 4, 7, 9, 12, 14, 17, 22, 23, 27, 31, 33, 37, 41, 45, 48, 53, 56, 59, 63, 67, 72, 77, 80, 83, 87, 90, 94, 103, 107, 111, 113, 121, 124, 128, 134, 138, 144, 148, 150, 158, 160, 164, 166, 175, 184, 188, 190, 193, 199, 201, 209, 214, 219, 226, 228, 234

Offset: 1

Odimar Fabeny, Aug 28 2014

			2 P 4 = 0,
3 4 P 6 = 1,
5 6 P 8 9 10 = 3,
7 8 9 10 P 12 P 14 = 4,
11 12 P 14 15 16 P 18 P 20 21 22 = 7
and so on.

Jens Kruse Andersen, Table of n, a(n) for n = 1..10000

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.

A246515 := proc(n) local p;  p:=ithprime(n); n - 1 + p - numtheory:-pi(2*p - 1); end; # N. J. A. Sloane, Oct 20 2024
[seq(A246515(n),n=1..120)];

Table[Prime[n] - PrimePi[2*Prime[n]] + n - 1, {n, 100}] (* Paolo Xausa, Oct 22 2024 *)

s=[]; forprime(p=2, 1000, n=0; for(q=p+1, 2*p-1, if(!isprime(q), n++)); s=concat(s, n)); s \\ Colin Barker, Aug 28 2014

a(n)=prime(n)+n-1-primepi(2*prime(n))
vector(100, n, a(n)) \\ Faster program. Jens Kruse Andersen, Aug 28 2014

from sympy import prime, primepi
def A246514(n): return (m:=prime(n))+n-1-primepi(m<<1) # Chai Wah Wu, Oct 22 2024

a(n) + A070046(n) = number of numbers between prime(n) and 2*prime(n), which is prime(n)-1. - N. J. A. Sloane, Aug 28 2014

More terms from Colin Barker, Aug 28 2014

A307912 a(n) = n - 1 - pi(2*n-1) + pi(n), where pi is the prime counting function.

Original entry on oeis.org

0, 0, 1, 1, 3, 3, 4, 5, 5, 5, 7, 7, 9, 10, 10, 10, 12, 13, 14, 15, 15, 15, 17, 17, 18, 19, 19, 20, 22, 22, 23, 24, 25, 25, 26, 26, 27, 28, 29, 29, 31, 31, 33, 34, 34, 35, 37, 38, 38, 39, 39, 39, 41, 41, 41, 42, 42, 43, 45, 46, 48, 49, 50, 50, 51, 51, 53, 54

Offset: 1

Wesley Ivan Hurt, May 09 2019

For n > 1, a(n) is the number of composites in the closed interval [n+1, 2n-1].

a(n) is also the number of composites appearing among the largest parts of the partitions of 2n into two distinct parts.

			a(7) = 4; there are 4 composites in the closed interval [8, 13]: 8, 9, 10 and 12.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Index entries for sequences related to partitions

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.

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855
A307912 := proc(n) chi(2*n-1) - chi(n); end;
A := [seq(A307912(n),n=1..120)]; # N. J. A. Sloane, Oct 20 2024

Table[n - 1 - PrimePi[2 n - 1] + PrimePi[n], {n, 100}]

from sympy import primepi
def A307912(n): return n+primepi(n)-primepi((n<<1)-1)-1 # Chai Wah Wu, Oct 20 2024

a(n) = n - 1 - A060715(n).

a(n) = n - 1 - A000720(2*n-1) + A000720(n).

A307989 a(n) = n - pi(2*n) + pi(n-1), where pi is the prime counting function.

Original entry on oeis.org

0, 0, 1, 2, 3, 4, 4, 6, 6, 6, 7, 8, 9, 11, 11, 11, 12, 14, 14, 16, 16, 16, 17, 18, 19, 20, 20, 21, 22, 23, 23, 25, 26, 26, 27, 27, 27, 29, 30, 30, 31, 32, 33, 35, 35, 36, 37, 39, 39, 40, 40, 40, 41, 42, 42, 43, 43, 44, 45, 47, 48, 50, 51, 51, 52, 52, 53, 55

Offset: 1

Wesley Ivan Hurt, May 09 2019

a(n) is the number of composites in the closed interval [n, 2n-1].

a(n) is also the number of composites among the largest parts of the partitions of 2n into two parts.

			a(7) = 4; There are 7 partitions of 2*7 = 14 into two parts (13,1), (12,2), (11,3), (10,4), (9,5), (8,6), (7,7). Among the largest parts 12, 10, 9 and 8 are composite, so a(7) = 4.

N. J. A. Sloane, Table of n, a(n) for n = 1..20000
Index entries for sequences related to partitions

Cf. A000720, A035250.
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.

chi := proc(n) if n <= 3 then 0 else n - numtheory:-pi(n) - 1; fi; end; # A065855
A307989 := proc(n) chi(2*n-1) - chi(n-1); end;
a := [seq(A307989(n),n=1..120)];

Table[n - PrimePi[2 n] + PrimePi[n - 1], {n, 100}]

from sympy import primepi
def A307989(n): return n+primepi(n-1)-primepi(n<<1) # Chai Wah Wu, Oct 20 2024

a(n) = n - A035250(n).

a(n) = n - A000720(2*n) + A000720(n-1).

cp's OEIS Frontend

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

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A056171 a(n) = pi(n) - pi(floor(n/2)), where pi is A000720.

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A070046 Number of primes between prime(n) and 2*prime(n) exclusive.

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A063124 a(n) = # { primes p | prime(n) <= p < 2*prime(n) } where prime(n) is the n-th prime.

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A075084 Number of composite numbers c with n <= c <= 2*n.

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A077463 Number of primes p such that n < p < 2n-2.

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