

A060199


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


15



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
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OFFSET

0,2


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
CullyHugill shows the above for n > exp(exp(32.892)) = 6.92619... * 10^83675518094285.  Charles R Greathouse IV, Aug 02 2021


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, 532562.
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. 117153. arXiv:0810.2113 [math.NT], 20082013.
Michaela CullyHugill, 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. 177197. 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), 255266.
MacTutor, A. E. Ingham Biography


FORMULA

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


EXAMPLE

n = 2: there are 5 primes between 8 and 27, 11,13,17,19,23.
n = 9, n+1 = 10: PrimePi(1000)PrimePi(729) = 168129 = a(9) = 39.


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 *)


PROG

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


CROSSREFS

First differences of A038098.
Cf. A000720, A014085, A014220, A061235, A062517.
Sequence in context: A125603 A239231 A078507 * A229240 A322135 A034705
Adjacent sequences: A060196 A060197 A060198 * A060200 A060201 A060202


KEYWORD

nonn


AUTHOR

Labos Elemer, Mar 19 2001


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


STATUS

approved



