A065478 Decimal expansion of Product_{p prime} (1 - p/(p^3-1)).

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

Offset: 0

Author

N. J. A. Sloane, Nov 19 2001

Keywords

• cons
• nonn

Examples

			0.57595996889294543964316337549249669...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, p. 106.
• Paulo Ribenboim, My Numbers, My Friends: Popular Lectures on Number Theory, Springer-Verlag, NY, 2000, p. 16.

Links

• Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 155.
• Yilan Hu and Carl Pomerance, The average order of elements in the multiplicative group of a finite field, involve, Vol. 5 (2012), No. 2, 229-236. See p. 8.
• Sungjin Kim, Average results on the order of a modulo p, Journal of Number Theory, Vol. 169 (2016), pp. 353-368; arXiv preprint, arXiv:1509.01752 [math.NT], 2015.
• Pär Kurlberg and Carl Pomerance, On a problem of Arnold: the average multiplicative order of a given integer, Algebra and Number Theory, Vol. 7, No. 4 (2013), pp. 981-999; alternative link.
• Pieter Moree and Peter Stevenhagen, A two-variable Artin conjecture, Journal of Number Theory, Vol. 85, No. 2 (2000), pp. 291-304; arXiv preprint, arXiv:math/9912250 [math.NT], 1999.
• G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
• P. J. Stephens, An average result for Artin's conjecture, Mathematika, Vol. 16, No. 2 (1969), pp. 178-188.
• P. J. Stephens, Prime divisors of second-order linear recurrences. I,, Journal of Number Theory, Vol. 8, No. 3 (1976), pp. 313-332.
• Eric Weisstein's World of Mathematics, Stephens' Constant.
• Wikipedia, Stephens' constant.

Crossrefs

Cf. A078079.

Programs

Mathematica

$MaxExtraPrecision = 100; m0 = 200; dm = 200; digits = 101; Clear[f]; f[m_] := f[m] = (slog = Normal[Series[Log[1 - p/(p^3 - 1)], {p, Infinity, m}]]; Exp[slog] /. Power[p, n_] -> PrimeZetaP[-n] // N[#, digits+10]&); f[m = m0]; Print[m, " ", f[m]]; f[m = m + dm]; While[Print[m, " ", f[m]]; RealDigits[f[m], 10, digits+5] != RealDigits[f[m - dm], 10, digits+5], m = m + dm];RealDigits[f[m], 10, digits] // First (* Jean-François Alcover, Sep 15 2015 *)

PARI

prodeulerrat(1 - p/(p^3-1)) \\ Amiram Eldar, Mar 17 2021