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A060199
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Number of primes between n^3 and (n+1)^3.
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8
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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; internal format)
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OFFSET
| 0,2
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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 this sequence is infinite. 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.
Conjecture: There are always more than 3 primes between two consecutive non 0 cubes. - Cino Hilliard (hillcino368(AT)gmail.com), Jan 05 2003
Cheng shows that a(n) > 0 for n > exp(exp(15)) = 1.607... * 10^1419716. [Charles R Greathouse IV, Nov 28 2011]
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REFERENCES
| 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. [From T. D. Noe (noe(AT)sspectra.com), Sep 27 2008]
A. E. Ingham, On the difference between two consecutive primes, Quart. J. Math. Oxford 8 (1937), 255-266.
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LINKS
| Charles Greathouse IV, Table of n, a(n) for n = 0..10000
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
A. E. Ingham Biography
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].
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FORMULA
| Table[PrimePi[(j+1)^3]-PrimePi[j^3], {j, 1, 100}]
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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
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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 (hillcino368(AT)gmail.com), Jan 05 2003
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CROSSREFS
| Cf. A000720, A014085.
Sequence in context: A010405 A125603 A078507 * A034705 A006844 A022425
Adjacent sequences: A060196 A060197 A060198 * A060200 A060201 A060202
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KEYWORD
| nonn
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AUTHOR
| Labos E. (labos(AT)ana.sote.hu), Mar 19 2001
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EXTENSIONS
| Corrected and added more detail to the Ingham references. - T. D. Noe (noe(AT)sspectra.com), Sep 23 2008
Combined two comments, correcting a bad error in the first comment. - T. D. Noe (noe(AT)sspectra.com), Sep 27 2008
Edited by N. J. A. Sloane (njas(AT)research.att.com), Jan 17 2009 at the suggestion of R. J. Mathar
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