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A108739 Mills' constant A generates a sequence of primes via b(n)= floor(A^3^n). This sequence is a(n) = b(n+1)-b(n)^3. 6
3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220, 66768, 300840, 1623568 (list; graph; refs; listen; history; text; internal format)



This allows larger terms of A051254 (which triple in digits each entry) to be given. Like A051254, currently requires Riemann Hypothesis to show sequence continues.

Currently a(11)=66768 generates only a probable prime number. - Arkadiusz Wesolowski, May 28 2011

Likewise a(12) and a(13) generate only a probable prime numbers, as well as being conditional on a(11) and a(12) being proved primes. Minimality of a(12)-a(13) is exhaustively tested. - Serge Batalov, Aug 06 2013


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


Table of n, a(n) for n=1..13.

Chris K. Caldwell, Mills' Theorem - a generalization.

Chris K. Caldwell, The List of Largest Known Primes, The 11th Mills' prime

Chris K. Caldwell and Yuanyou Cheng, Determining Mills' Constant and a Note on Honaker's Problem, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.1.

Henri & Renaud Lifchitz, PRP Records

W. H. Mills, A prime-representing function, Bull. Amer. Math. Soc., Vol. 53 (1947), p. 604.

Eric Weisstein's World of Mathematics, Mills' Constant

Eric Weisstein's World of Mathematics, Mills' Prime

E. M. Wright, A class of representing functions, J. London Math. Soc., Vol. 29 (1954) pp. 63-71.


b(1) = 2; b(n+1) = nextprime(b(n)^3); a(n) = b(n+1)-b(n)^3;


The Mills' primes (given in A051254) are 2, 2^3+3 = 11, (2^3+3)^3+30 = 11^3+30 = 1361, ((2^3+3)^3+30)^3+6 = 1361^3+6 = 2521008887, etc. The terms added at each step yield this sequence. They are the least positive integers which added to the cube of the preceding prime yield again a prime, cf. formula. - M. F. Hasler, Jul 22 2013


Cf. A051254, A051021.

Sequence in context: A318971 A082792 A078242 * A072973 A154054 A118219

Adjacent sequences:  A108736 A108737 A108738 * A108740 A108741 A108742




Chris K. Caldwell, Jun 22 2005


a(9)-a(11) from Caldwell and Cheng, Aug 29 2005

Corrected by T. D. Noe, Sep 24 2008

a(12) (which generates a PRP) from Serge Batalov, Jul 19 2013

a(13) (which generates a PRP) from Serge Batalov, Aug 06 2013



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Last modified November 12 19:19 EST 2018. Contains 317116 sequences. (Running on oeis4.)