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

Author

Labos Elemer, Mar 19 2001

Keywords

• nonn

Comments

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

Examples

			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.
		

Links

• 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]

Crossrefs

First differences of A038098.

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

Programs

Magma

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

Mathematica

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 *)

PARI

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

Python

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

Formula

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

Extensions

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