A038098 -id:A038098 - cp's OEIS frontend

A060199 Number of primes between n^3 and (n+1)^3.

0, 4, 5, 9, 12, 17, 21, 29, 32, 39, 49, 52, 58, 73, 76, 88, 92, 109, 117, 125, 140, 151, 159, 176, 188, 199, 207, 233, 247, 254, 267, 284, 305, 320, 346, 338, 373, 385, 416, 418, 437, 458, 481, 504, 517, 551, 555, 583, 599, 636, 648, 678, 686, 733, 723, 753, 810

Offset: 0

Labos Elemer, Mar 19 2001

nonn

Ingham showed that for n large enough and k=5/8, prime(n+1)-prime(n) = O(prime(n)^k). Ingham's result implies that there is a prime between sufficiently large consecutive cubes. Therefore a(n) is nonzero for n sufficiently large. Using the Riemann Hypothesis, Caldwell and Cheng prove there is a prime between all consecutive cubes. The question is undecided for squares. Many authors have reduced the value of k. The best value of k is 21/40, proved by Baker, Harman and Pintz in 2001. - corrected by Jonathan Sondow, May 19 2013
Conjecture: There are always more than 3 primes between two consecutive nonzero cubes. - Cino Hilliard, Jan 05 2003
Dudek (2014), correcting a claim of Cheng, shows that a(n) > 0 for n > exp(exp(33.217)) = 3.06144... * 10^115809481360808. - Charles R Greathouse IV, Jun 27 2014
Cully-Hugill shows the above for n > exp(exp(32.892)) = 6.92619... * 10^83675518094285. - Charles R Greathouse IV, Aug 02 2021
Mossinghoff, Trudgian, & Yang improve this to n > exp(exp(32.76)) = 3.62275 * 10^73328286790528. - Charles R Greathouse IV, Jul 31 2024

			n = 2: there are 5 primes between 8 and 27, 11,13,17,19,23.
n = 9, n+1 = 10: PrimePi(1000)-PrimePi(729) = 168-129 = a(9) = 39.

Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
R. C. Baker, G. Harman and J. Pintz, The difference between consecutive primes, II, Proc. London Math. Soc. (3) 83 (2001), no. 3, 532-562.
Chris K. Caldwell and Yuanyou Cheng, Determining Mills's Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.
Y.-Y. F.-R. Cheng, Explicit Estimate on Primes between consecutive cubes, Rocky Mountain Journal of Mathematics 40:1 (2010), pp. 117-153. arXiv:0810.2113 [math.NT], 2008-2013.
Michaela Cully-Hugill, Primes between consecutive powers, arXiv:2107.14468 [math.NT]
Adrian Dudek, An explicit result for primes between cubes arXiv:1401.4233 [math.NT], 2014.
Adrian Dudek, An explicit result for primes between cubes, Functiones et Approximatio Commentarii Mathematici Vol. 55, Issue 2 (Dec 2016), pp. 177-197. See also Explicit Estimates in the Theory of Prime Numbers, arXiv:1611.07251 [math.NT], 2016; PhD thesis, Australian National University, 2016.
A. E. Ingham, On the difference between consecutive primes, Quart. J. Math. Oxford 8 (1937), 255-266.
MacTutor, A. E. Ingham Biography
Michael J. Mossinghoff, Timothy S. Trudgian, and Andrew Yang, Explicit zero-free regions for the Riemann zeta-function, arXiv preprint (2022). arXiv:2212.06867 [math.NT]

First differences of A038098.

Cf. A000720, A014085, A014220, A061235, A062517.

[0] cat [#PrimesInInterval(n^3, (n+1)^3): n in [1..70]]; // Vincenzo Librandi, Feb 13 2016

PrimePi[(#+1)^3]-PrimePi[#^3]&/@Range[0,60] (* Harvey P. Dale, Feb 08 2013 *)
Last[#]-First[#]&/@Partition[PrimePi[Range[0,60]^3],2,1] (* Harvey P. Dale, Feb 02 2015 *)

cubespr(n)= for(x=0,n, ct=0; for(y=x^3,(x+1)^3, if(isprime(y), ct++; )); if(ct>=0,print1(ct, ", ")))  \\ Cino Hilliard, Jan 05 2003

from sympy import primepi
def a(n): return primepi((n+1)**3) - primepi(n**3)
print([a(n) for n in range(57)]) # Michael S. Branicky, Jun 22 2021

Table[PrimePi[(j+1)^3]-PrimePi[j^3], {j, 1, 100}]

Corrected and added more detail to the Ingham references. - T. D. Noe, Sep 23 2008
Combined two comments, correcting a bad error in the first comment. - T. D. Noe, Sep 27 2008
Edited by N. J. A. Sloane, Jan 17 2009 at the suggestion of R. J. Mathar

A086688 Number of primes < prime(n)^3.

Original entry on oeis.org

4, 9, 30, 68, 217, 327, 656, 882, 1457, 2707, 3228, 5194, 6850, 7789, 9916, 13746, 18411, 20179, 26062, 30597, 32984, 40991, 46959, 56924, 72217, 80725, 85182, 94695, 99669, 110127, 152280, 165951, 187910, 195660, 237427, 246434, 274614

Offset: 1

Cino Hilliard, Jul 28 2003

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000720, A030078, A038098.

a[n_] := PrimePi[Prime[n]^3]; Array[a, 50] (* Amiram Eldar, Sep 06 2024 *)

a(n) = primepi(prime(n)^3); \\ Michel Marcus, Sep 18 2023

a(n) = A038098(prime(n)). - R. J. Mathar, Sep 15 2012

a(n) = A000720(A030078(n)). - Michel Marcus, Sep 18 2023

A060304 Number of primes below n^3 does not exceed n times the number of primes below n^2.

Original entry on oeis.org

0, 0, 3, 6, 15, 19, 37, 47, 69, 82, 113, 139, 180, 216, 244, 300, 381, 423, 486, 553, 638, 726, 820, 887, 1029, 1152, 1256, 1376, 1527, 1659, 1794, 1992, 2156, 2357, 2517, 2739, 2909, 3085, 3365, 3627, 3933, 4200, 4380, 4687, 4960, 5313, 5547, 5917, 6395

Offset: 0

Labos Elemer, Mar 26 2001

nonn

			n=10, 10*pi(100)=250, pi(1000)=168, a(10)=250-168=82.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
S. Segal, On π(x+y)<=π(x)+π(y), Transactions American Mathematical Society, 104 (1962), 523-527.

Cf. A060199, A060208, A000720.

Table[n*PrimePi[n^2]-PrimePi[n^3], {n, 1, 100}]

a(n) = n*pi(n*n) - pi(n*n*n). - Jonathan Sondow, Feb 17 2014

a(n) = n*A038107(n) - A038098(n). - Michel Marcus, Feb 17 2014

A365767 a(n) is the number of primes between (prime(n))^3 and (prime(n+1))^3.

Original entry on oeis.org

5, 21, 38, 149, 110, 329, 226, 575, 1250, 521, 1966, 1656, 939, 2127, 3830, 4665, 1768, 5883, 4535, 2387, 8007, 5968, 9965, 15293, 8508, 4457, 9513, 4974, 10458, 42153, 13671, 21959, 7750, 41767, 9007, 28180, 30226, 21322, 33813, 35899, 12506, 66241, 14003, 28809, 14848, 94735

Offset: 1

Zak Seidov and Robert Israel, Sep 18 2023

nonn

			a(2) = 21 because there are 21 primes between prime(2)^3 = 27 and prime(3)^3 = 125.

First differences of A086688.

Cf. A038098, A050216.

A:= [seq(numtheory:-pi(ithprime(i)^3),i=1..30)]:
A[2..-1] - A[1..-2];

a(n) = A038098(A000040(n+1)) - A038098(A000040(n)).

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A060199 Number of primes between n^3 and (n+1)^3.

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A086688 Number of primes < prime(n)^3.

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A060304 Number of primes below n^3 does not exceed n times the number of primes below n^2.

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A365767 a(n) is the number of primes between (prime(n))^3 and (prime(n+1))^3.

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