




OFFSET

1,1


COMMENTS

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).
Obverse of this is A118910 a(1) = 2; a(n) is greatest prime < a(n1)^3.  Jonathan Vos Post, May 05 2006


REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, SpringerVerlag, 1976, page 8.


LINKS

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.
S. R. Finch, Mills' Constant
James Grime and Brady Haran, Awesome Prime Number Constant, Numberphile video, 2013.
Brian Hayes, Pumping the Primes, bitplayer, Aug 19 2015.
W. H. Mills, A primerepresenting function, Bull. Amer. Math. Soc., Vol. 53 (1947), p. 604.
László Tóth, A Variation on MillsLike PrimeRepresenting Functions, arXiv:1801.08014 [math.NT], 2018.
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


FORMULA

a(1) = 2; a(n) is least prime > a(n1)^3.  Jonathan Vos Post, May 05 2006


EXAMPLE

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
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


MAPLE

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


MATHEMATICA

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 *)


PROG

(PARI) a(n)=if(n==1, 2, nextprime(a(n1)^3)) \\ Charles R Greathouse IV, Jun 23 2017


CROSSREFS

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


KEYWORD

nonn,nice


AUTHOR

Simon Plouffe.


EXTENSIONS

Edited by N. J. A. Sloane, May 05 2007


STATUS

approved



