A051021 Decimal expansion of Mills's constant, assuming the Riemann Hypothesis is true.

1, 3, 0, 6, 3, 7, 7, 8, 8, 3, 8, 6, 3, 0, 8, 0, 6, 9, 0, 4, 6, 8, 6, 1, 4, 4, 9, 2, 6, 0, 2, 6, 0, 5, 7, 1, 2, 9, 1, 6, 7, 8, 4, 5, 8, 5, 1, 5, 6, 7, 1, 3, 6, 4, 4, 3, 6, 8, 0, 5, 3, 7, 5, 9, 9, 6, 6, 4, 3, 4, 0, 5, 3, 7, 6, 6, 8, 2, 6, 5, 9, 8, 8, 2, 1, 5, 0, 1, 4, 0, 3, 7, 0, 1, 1, 9, 7, 3, 9, 5, 7, 0, 7, 2, 9

Offset: 1

Author

Eric W. Weisstein

Keywords

• nonn
• cons

Comments

Not known to be rational or irrational. See Saito (2024) for a new result. - Charles R Greathouse IV, Jul 18 2013, Hugo Pfoertner, May 01 2024

Examples

			1.3063778838630806904686144926026057129167845851567136443680537599664340537668...
		

References

• T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 8.
• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Section 2.13, p. 130.
• Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 137.

Links

• Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (first 641 terms from Tin Apato)
• C. K. Caldwell, Mills's Constant [Gives 6000 terms assuming the Riemann Hypothesis.]
• 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.
• Christian Elsholtz, Unconditional Prime-representing Functions, Following Mills, arXiv:2004.01285 [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.
• Aminu Alhaji Ibrahim and Sa’idu Isah Abubaka, Aunu Integer Sequence as Non-Associative Structure and Their Graph Theoretic Properties, Advances in Pure Mathematics, 2016, 6, 409-419.
• William H. Mills, A prime-representing function, Bull. Amer. Math. Soc., Vol. 53, No. 6 (1947), p. 604; Errata, ibid., Vol. 53, No 12 (1947), p. 1196.
• Bernard Montaron, Exponential prime sequences, arXiv:2011.14653 [math.NT], 2020.
• Robert P. Munafo, Notable Properties of Specific Numbers.
• Simon Plouffe, The calculation of p(n) and pi(n), arXiv:2002.12137 [math.NT], 2020.
• Kota Saito, Mills' constant is irrational, arXiv:2404.19461 [math.NT], 2024.
• László Tóth, A Variation on Mills-Like Prime-Representing Functions, arXiv:1801.08014 [math.NT], 2018.
• Eric Weisstein's World of Mathematics, Mills' Constant.
• Eric Weisstein's World of Mathematics, Prime Formulas.

Crossrefs

Cf. A051254.

Programs

Mathematica

RealDigits[ Nest[ NextPrime[#^3] &, 2, 7]^(1/3^8), 10, 111][[1]] (* Robert G. Wilson v, Nov 14 2012 *)

PARI

A051021_upto(N=99)=localprec(N+9);digits(10^N*sqrtn(A051254(N=logint(N,3)+2),3^N)\1) \\ M. F. Hasler, Sep 11 2024

Extensions

More terms from Robert G. Wilson v, Sep 08 2000

More terms from Tin Apato (tinapto(AT)yahoo.es), Dec 12 2007