|
%I #66 Sep 08 2022 08:45:03
%S 0,4,5,9,12,17,21,29,32,39,49,52,58,73,76,88,92,109,117,125,140,151,
%T 159,176,188,199,207,233,247,254,267,284,305,320,346,338,373,385,416,
%U 418,437,458,481,504,517,551,555,583,599,636,648,678,686,733,723,753,810
%N Number of primes between n^3 and (n+1)^3.
%C 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
%C Conjecture: There are always more than 3 primes between two consecutive nonzero cubes. - _Cino Hilliard_, Jan 05 2003
%C 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
%C Cully-Hugill shows the above for n > exp(exp(32.892)) = 6.92619... * 10^83675518094285. - _Charles R Greathouse IV_, Aug 02 2021
%H Charles R Greathouse IV, <a href="/A060199/b060199.txt">Table of n, a(n) for n = 0..10000</a>
%H R. C. Baker, G. Harman and J. Pintz, <a href="http://www.cs.umd.edu/~gasarch/BLOGPAPERS/BakerHarmanPintz.pdf">The difference between consecutive primes, II</a>, Proc. London Math. Soc. (3) 83 (2001), no. 3, 532-562.
%H Chris K. Caldwell and Yuanyou Cheng, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html">Determining Mills's Constant and a Note on Honaker's Problem</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.
%H Y.-Y. F.-R. Cheng, <a href="http://arxiv.org/abs/0810.2113">Explicit Estimate on Primes between consecutive cubes</a>, Rocky Mountain Journal of Mathematics 40:1 (2010), pp. 117-153. arXiv:0810.2113 [math.NT], 2008-2013.
%H Michaela Cully-Hugill, <a href="https://arxiv.org/abs/2107.14468">Primes between consecutive powers</a>, arXiv:2107.14468 [math.NT]
%H Adrian Dudek, <a href="http://arxiv.org/abs/1401.4233">An explicit result for primes between cubes</a> arXiv:1401.4233 [math.NT], 2014.
%H 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 <a href="https://arxiv.org/abs/1611.07251">Explicit Estimates in the Theory of Prime Numbers</a>, arXiv:1611.07251 [math.NT], 2016; PhD thesis, Australian National University, 2016.
%H A. E. Ingham, <a href="http://dx.doi.org/10.1093/qmath/os-8.1.255">On the difference between consecutive primes</a>, Quart. J. Math. Oxford 8 (1937), 255-266.
%H MacTutor, <a href="http://www-groups.dcs.st-and.ac.uk/~history/Mathematicians/Ingham.html">A. E. Ingham Biography</a>
%F Table[PrimePi[(j+1)^3]-PrimePi[j^3], {j, 1, 100}]
%e n = 2: there are 5 primes between 8 and 27, 11,13,17,19,23.
%e n = 9, n+1 = 10: PrimePi(1000)-PrimePi(729) = 168-129 = a(9) = 39.
%t PrimePi[(#+1)^3]-PrimePi[#^3]&/@Range[0,60] (* _Harvey P. Dale_, Feb 08 2013 *)
%t Last[#]-First[#]&/@Partition[PrimePi[Range[0,60]^3],2,1] (* _Harvey P. Dale_, Feb 02 2015 *)
%o (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
%o (Magma) [0] cat [#PrimesInInterval(n^3, (n+1)^3): n in [1..70]]; // _Vincenzo Librandi_, Feb 13 2016
%o (Python)
%o from sympy import primepi
%o def a(n): return primepi((n+1)**3) - primepi(n**3)
%o print([a(n) for n in range(57)]) # _Michael S. Branicky_, Jun 22 2021
%Y First differences of A038098.
%Y Cf. A000720, A014085, A014220, A061235, A062517.
%K nonn
%O 0,2
%A _Labos Elemer_, Mar 19 2001
%E Corrected and added more detail to the Ingham references. - _T. D. Noe_, Sep 23 2008
%E Combined two comments, correcting a bad error in the first comment. - _T. D. Noe_, Sep 27 2008
%E Edited by _N. J. A. Sloane_, Jan 17 2009 at the suggestion of _R. J. Mathar_
|
