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A051254 Mills primes. 15

%I

%S 2,11,1361,2521008887,16022236204009818131831320183,

%T 4113101149215104800030529537915953170486139623539759933135949994882770404074832568499

%N Mills primes.

%C 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).

%C Obverse of this is A118910 a(1) = 2; a(n) is greatest prime < a(n-1)^3. - _Jonathan Vos Post_, May 05 2006

%D T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.

%H Robert G. Wilson v, <a href="/A051254/b051254.txt">Table of n, a(n) for n = 1..8</a>

%H Chris K. Caldwell, <a href="http://www.utm.edu/research/primes/notes/proofs/A3n.html">Mills' Theorem - a generalization</a>

%H Chris K. Caldwell and Yuanyou Chen, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL8/Caldwell/caldwell78.html">Determining Mills' Constant and a Note on Honaker's Problem</a>, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.

%H Steven R. Finch, <a href="http://www.people.fas.harvard.edu/~sfinch/constant/mills/mills.html">Mills' Constant</a> [Broken link]

%H Steven R. Finch, <a href="http://web.archive.org/web/20010603070928/http://www.mathsoft.com/asolve/constant/mills/mills.html">Mills' Constant</a> [From the Wayback machine]

%H Dylan Fridman et al., <a href="https://www.researchgate.net/publication/330746181_A_Prime-Representing_Constant">A prime-representing constant</a>, Amer. Math. Monthly 126 (2019), 72-73 (on ResearchGate).

%H James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=6ltrPVPEwfo">Awesome Prime Number Constant</a>, Numberphile video, 2013.

%H Brian Hayes, <a href="http://bit-player.org/2015/pumping-the-primes">Pumping the Primes</a>, bit-player, Aug 19 2015.

%H W. H. Mills, <a href="http://dx.doi.org/10.1090/S0002-9904-1947-08849-2">A prime-representing function</a>, Bull. Amer. Math. Soc., Vol. 53 (1947), p. 604.

%H László Tóth, <a href="https://arxiv.org/abs/1801.08014">A Variation on Mills-Like Prime-Representing Functions</a>, arXiv:1801.08014 [math.NT], 2018.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MillsPrime.html">Mills' Prime</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeFormulas.html">Prime Formulas</a>

%H Eric W. Weisstein, <a href="/A051254/a051254.txt">Table of n, a(n) for n = 1..13</a>

%F a(1) = 2; a(n) is least prime > a(n-1)^3. - _Jonathan Vos Post_, May 05 2006

%e 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

%e a(4) = 16022236204009818131831320183 = a(3)^3 + 80 = 2521008887^3 + 80 and there is no smaller k such that a(3)^3 + k is prime. - _Jonathan Vos Post_, May 05 2006

%p floor(A^(3^n), n=1..10); # A is Mills's constant: 1.306377883863080690468614492602605712916784585156713644368053759966434.. (A051021).

%t p = 1; Table[p = NextPrime[p^3], {6}] (* _T. D. Noe_, Sep 24 2008 *)

%t NestList[NextPrime[#^3] &, 2, 5] (* _Harvey P. Dale_, Feb 28 2012 *)

%o (PARI) a(n)=if(n==1, 2, nextprime(a(n-1)^3)) \\ _Charles R Greathouse IV_, Jun 23 2017

%Y Cf. A001358, A055496, A076656, A006992, A005384, A005385, A118908, A118909, A118910, A118911, A118912, A118913.

%Y Cf. A224845 (integer lengths of Mills primes).

%Y Cf. A108739 (sequence of offsets b_n associated with Mills primes).

%Y Cf. A051021 (decimal expansion of Mills constant).

%K nonn,nice

%O 1,1

%A _Simon Plouffe_.

%E Edited by _N. J. A. Sloane_, May 05 2007

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Last modified June 20 09:12 EDT 2019. Contains 324234 sequences. (Running on oeis4.)