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 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 (list; graph; refs; listen; history; text; internal format)
 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 Cully-Hugill 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, 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 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) = 168-129 = 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)  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

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Last modified February 2 10:36 EST 2023. Contains 360011 sequences. (Running on oeis4.)