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OFFSET
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1,1
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COMMENTS
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Mills showed that there is a number A > 1 but not an integer, such that floor( A^(3^n) ) is a prime for all n = 1, 2, 3, ... A is approximately 1.306377883863... (see A051021).
a(1) = 2 and (for n > 1) a(n) is greatest prime < a(n-1)^3. - Jonathan Vos Post, May 05 2006
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.
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LINKS
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Robert G. Wilson v, Table of n, a(n) for n = 1..8
Chris K. Caldwell, Mills' Theorem - a generalization
Chris K. Caldwell and Yuanyou Chen, Determining Mills' Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.
Steven R. Finch, Mills' Constant [Broken link]
Steven R. Finch, Mills' Constant [From the Wayback machine]
Dylan Fridman et al., A prime-representing constant, Amer. Math. Monthly 126 (2019), 72-73 (on ResearchGate); also on arXiv, arXiv:2010.15882 [math.NT], 2020.
James Grime and Brady Haran, Awesome Prime Number Constant, Numberphile video, 2013.
Brian Hayes, Pumping the Primes, bit-player, Aug 19 2015.
W. H. Mills, A prime-representing function, Bull. Amer. Math. Soc., Vol. 53 (1947), p. 604.
Simon Plouffe, The calculation of p(n) and pi(n), arXiv:2002.12137 [math.NT], 2020.
László Tóth, A Variation on Mills-Like Prime-Representing Functions, arXiv:1801.08014 [math.NT], 2018.
Juan L. Varona, A Couple of Transcendental Prime-Representing Constants, arXiv:2012.11750 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Mills' Prime
Eric Weisstein's World of Mathematics, Prime Formulas
Eric W. Weisstein, Table of n, a(n) for n = 1..13
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FORMULA
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a(1) = 2; a(n) is least prime > a(n-1)^3. - Jonathan Vos Post, May 05 2006
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EXAMPLE
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a(3) = 1361 = 11^3 + 30 = a(2)^3 + 30 and there is no smaller k such that a(2)^3 + k is prime. - Jonathan Vos Post, May 05 2006
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MAPLE
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floor(A^(3^n), n=1..10); # A is Mills's constant: 1.306377883863080690468614492602605712916784585156713644368053759966434.. (A051021).
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MATHEMATICA
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p = 1; Table[p = NextPrime[p^3], {6}] (* T. D. Noe, Sep 24 2008 *)
NestList[NextPrime[#^3] &, 2, 5] (* Harvey P. Dale, Feb 28 2012 *)
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PROG
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(PARI) a(n)=if(n==1, 2, nextprime(a(n-1)^3)) \\ Charles R Greathouse IV, Jun 23 2017
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CROSSREFS
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Cf. A001358, A055496, A076656, A006992, A005384, A005385, A118908, A118909, A118910, A118911, A118912, A118913.
Cf. A224845 (integer lengths of Mills primes).
Cf. A108739 (sequence of offsets b_n associated with Mills primes).
Cf. A051021 (decimal expansion of Mills constant).
Sequence in context: A034388 A131316 A062636 * A095820 A101295 A131306
Adjacent sequences: A051251 A051252 A051253 * A051255 A051256 A051257
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KEYWORD
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nonn,nice
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AUTHOR
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Simon Plouffe.
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EXTENSIONS
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Edited by N. J. A. Sloane, May 05 2007
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STATUS
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approved
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