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.

3, 30, 6, 80, 12, 450, 894, 3636, 70756, 97220, 66768, 300840, 1623568

Offset

1,1

Comments

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

References

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

Links

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.

Formula

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

Example

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

Mathematica

B[1] = 2; B[n_] := B[n] = NextPrime[B[n - 1]^3]; Table[B[n + 1] - B[n]^3, {n, 7}] (* Robert Price, Jun 09 2019 *)

Prog

(PARI) p=2; until(, np=nextprime(p^3); print1(np-p^3, ", "); p=np) \\ Jeppe Stig Nielsen, Apr 22 2020

Crossrefs

Cf. A051254, A051021.

Sequence in context: A318971 A082792 A078242 * A072973 A154054 A118219

Adjacent sequences: A108736 A108737 A108738 * A108740 A108741 A108742

Keyword

more,nonn,hard

Author

Chris K. Caldwell, Jun 22 2005

Extensions

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

Status

approved