A035250 -id:A035250 - cp's OEIS frontend

A289495 Number of primes in the interval [4n, 5n].

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

Offset: 1

FUNG Cheok Yin, Jul 07 2017

FUNG Cheok Yin, Table of n, a(n) for n = 1..10000

Cf. A035250, A289493, A289494, A289496, A289497, A289498, A289499, A289500.

a(n) = primepi(5*n) - primepi(4*n); \\ Michel Marcus, Jul 08 2017

A289496 Number of primes in the interval [5n, 6n].

Original entry on oeis.org

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

Offset: 1

FUNG Cheok Yin, Jul 07 2017

FUNG Cheok Yin, Table of n, a(n) for n = 1..10000

Cf. A035250, A289493, A289494, A289495, A289497, A289498, A289499, A289500.

Table[Count[Range[5n,6n],?PrimeQ],{n,80}] (* _Harvey P. Dale, Sep 07 2017 *)

A289497 Number of primes in the interval [6n, 7n].

Original entry on oeis.org

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

Offset: 1

FUNG Cheok Yin, Jul 07 2017

FUNG Cheok Yin, Table of n, a(n) for n = 1..10000

Cf. A035250, A289493, A289494, A289495, A289496, A289498, A289499, A289500.

a(n) = primepi(7*n) - primepi(6*n); \\ Michel Marcus, Jul 08 2017

A289498 Number of primes in the interval [7n, 8n].

Original entry on oeis.org

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

Offset: 1

FUNG Cheok Yin, Jul 07 2017

FUNG Cheok Yin, Table of n, a(n) for n = 1..10000

Cf. A035250, A289493, A289494, A289495, A289496, A289497, A289499, A289500.

A289499 Number of primes in the interval [8n, 9n].

Original entry on oeis.org

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

Offset: 1

FUNG Cheok Yin, Jul 07 2017

FUNG Cheok Yin, Table of n, a(n) for n = 1..10000

Cf. A035250, A289493, A289494, A289495, A289496, A289497, A289498, A289500.

A289500 Number of primes in the interval [9n, 10n].

Original entry on oeis.org

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

Offset: 1

FUNG Cheok Yin, Jul 12 2017

FUNG Cheok Yin, Table of n, a(n) for n = 1..10000

Cf. A035250, A289493, A289494, A289495, A289496, A289497, A289498, A289499.

Cf. A038801.

[0] cat [#PrimesInInterval(9*n, 10*n): n in [2..100]]; // Vincenzo Librandi, Jul 13 2017

seq(numtheory:-pi(10*n)-numtheory:-pi(9*n),n=1..100); # Robert Israel, Jul 12 2017

Join[{0}, Table[PrimePi[10 n] - PrimePi[9 n], {n, 2, 100}]] (* Vincenzo Librandi, Jul 13 2017 *)

a(n) = primepi(10*n) - primepi(9*n); \\ Michel Marcus, Jul 12 2017

a(n) = n/log(n) + (1 + log(3^18/10^10))*n/log(n)^2 + O(n/log(n)^3) as n -> infinity. - Robert Israel, Jul 12 2017

A099802 Bisection of A000720.

Original entry on oeis.org

1, 2, 3, 4, 4, 5, 6, 6, 7, 8, 8, 9, 9, 9, 10, 11, 11, 11, 12, 12, 13, 14, 14, 15, 15, 15, 16, 16, 16, 17, 18, 18, 18, 19, 19, 20, 21, 21, 21, 22, 22, 23, 23, 23, 24, 24, 24, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 30, 30, 30, 30, 30, 30, 31, 31, 32, 32, 32, 33, 34, 34, 34, 34, 34

Offset: 1

N. J. A. Sloane, Nov 19 2004

Maximal number of primes possible in a string of 2n consecutive numbers. - Lekraj Beedassy, Dec 04 2004
a(n) = A139325(n,1) + 1. - Reinhard Zumkeller, Apr 14 2008
Or the number of primes <= 2n. - Juri-Stepan Gerasimov, Oct 29 2009

Cf. A099081.

with(numtheory):seq(pi(2*n),n=1..86); # Emeric Deutsch, Apr 12 2005

a(n)=primepi(2*n) \\ Charles R Greathouse IV, Sep 02 2015

[prime_pi(2*n) for n in range(1, 75)] # Zerinvary Lajos, Jun 06 2009

a(n) = A000720(n) + A035250(n) - A010051(n). - Reinhard Zumkeller, Jul 05 2010

a(n) = A000720(2*n). - Wesley Ivan Hurt, Jun 16 2013

More terms from Emeric Deutsch, Apr 12 2005

A352749 a(n) = pi(n) * (pi(2n-1) - pi(n-1)).

Original entry on oeis.org

0, 2, 4, 4, 6, 6, 12, 8, 12, 16, 20, 20, 24, 18, 24, 30, 35, 28, 40, 32, 40, 48, 54, 54, 54, 54, 63, 63, 70, 70, 88, 77, 77, 88, 88, 99, 120, 108, 108, 120, 130, 130, 140, 126, 140, 140, 150, 135, 150, 150, 165, 180, 192, 192, 208, 208, 224, 224, 238, 221, 234, 216, 216, 234

Offset: 1

Wesley Ivan Hurt, Apr 01 2022

Number of ordered pairs of prime numbers, (p,q), such that p <= n <= q < 2n.

Also the number of ordered pairs of prime numbers, (p,q) that can be made with p <= q, where p and q appear as the smaller and larger parts (respectively) of the partitions of 2n into 2 parts that contain at least 1 prime.

			a(5) = 6; there are 6 ordered pairs of prime numbers, (p,q), such that p <= 5 <= q < 10: (2,5), (2,7), (3,5), (3,7), (5,5), and (5,7).
Another interpretation for a(5): the 3 partitions of 2*5 = 10 into 2 parts containing at least one prime are 2+8 = 3+7 = 5+5. There are 6 ordered pairs of primes (p,q) that can be made with p <= q, which are the same ordered pairs in the previous example.

Eric Weisstein's World of Mathematics, Prime Counting Function.
Wikipedia, Prime-counting function.

Cf. A000720 (pi), A035250, A352753, A352754.

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

a(n) = primepi(n)*(primepi(2*n-1) - primepi(n-1)); \\ Michel Marcus, Apr 01 2022

a(n) = Sum_{p <= n <= q < 2n, p,q prime} 1.

a(n) = A000720(n) * A035250(n). - Bernard Schott, Apr 02 2022

A059316 Least integer m such that between m and 2m (including endpoints) there are exactly n primes.

Original entry on oeis.org

1, 2, 7, 10, 16, 22, 27, 31, 36, 37, 51, 52, 55, 57, 70, 79, 87, 91, 96, 97, 100, 120, 121, 126, 135, 136, 142, 147, 157, 175, 177, 187, 190, 205, 210, 211, 217, 220, 222, 232, 246, 250, 255, 262, 289, 297, 300, 301, 304, 307, 310, 324, 327, 330, 331, 342, 346

Offset: 1

Felice Russo, Jan 25 2001

See A060756 for the case they are excluded. - R. J. Mathar, Nov 28 2007

A035250(a(n)) = n and A035250(m) <> n for m < a(n). - Reinhard Zumkeller, Jan 08 2012

			a(3)=7 because 7 is the least integer such that between 7 and 14 there are 3 primes.

T. D. Noe, Table of n, a(n) for n=1..1000
A related page
Wilkinson, Erdos' proof of Bertrand's postulate, MathForum(AT)Drexel.

Cf. A035250, A060756.

import Data.List (elemIndex)
import Data.Maybe (mapMaybe)
a059316 n = a059316_list !! n
a059316_list = map (+ 1) $ mapMaybe (`elemIndex` a035250_list) [1..]
-- Reinhard Zumkeller, Jan 05 2012

im[n_]:=Module[{m=1},While[PrimePi[2m]-(PrimePi[m-1])!=n,m++];m]; Array[  im,60]  (* Harvey P. Dale, May 19 2012 *)

A352753 a(n) = (pi(2n-1) - pi(n-1)) * Sum_{p <= n, p prime} p.

Original entry on oeis.org

0, 4, 10, 10, 20, 20, 51, 34, 51, 68, 112, 112, 164, 123, 164, 205, 290, 232, 385, 308, 385, 462, 600, 600, 600, 600, 700, 700, 903, 903, 1280, 1120, 1120, 1280, 1280, 1440, 1970, 1773, 1773, 1970, 2380, 2380, 2810, 2529, 2810, 2810, 3280, 2952, 3280, 3280, 3608

Offset: 1

Wesley Ivan Hurt, Apr 01 2022

nonn

Sum of the primes p from the ordered pairs of prime numbers, (p,q), such that p <= n <= q < 2n.

			a(5) = 20; there are 6 ordered pairs of prime numbers, (p,q), such that p <= 5 <= q < 10: (2,5), (2,7), (3,5), (3,7), (5,5), and (5,7). The sum of the corresponding prime parts p gives 2+2+3+3+5+5 = 20.

Cf. A000720 (pi), A034387, A035250, A352749, A352754, A352775, A352777.

Table[(PrimePi[2 n - 1] - PrimePi[n - 1]) Sum[k (PrimePi[k] - PrimePi[k - 1]), {k, n}], {n, 100}]

a(n) = A035250(n) * A034387(n). - Bernard Schott, Apr 02 2022

a(n) = A352775(n) - A352754(n).

cp's OEIS Frontend

A289495 Number of primes in the interval [4n, 5n].

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A289496 Number of primes in the interval [5n, 6n].

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A289497 Number of primes in the interval [6n, 7n].

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A289498 Number of primes in the interval [7n, 8n].

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A289499 Number of primes in the interval [8n, 9n].

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A289500 Number of primes in the interval [9n, 10n].

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A099802 Bisection of A000720.

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A352749 a(n) = pi(n) * (pi(2n-1) - pi(n-1)).

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A059316 Least integer m such that between m and 2m (including endpoints) there are exactly n primes.

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