A271798 Decimal expansion of Matthews' constant C_2, an analog of Artin's constant for primitive roots.

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

Offset

0,2

References

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.4 Artin's constant, p. 105.

Links

Table of n, a(n) for n=0..105.

K. R. Matthews, A generalisation of Artin's conjecture for primitive roots, Acta arithmetica, Vol. 29, No. 2 (1976), pp. 113-146.

Formula

C_2 = Product_{p prime} 1 - (p^2 - (p - 1)^2)/(p^2*(p - 1)).

Log(1 - (p^2 - (p - 1)^2)/(p^2*(p - 1))) + O(p,Infinity)^m = Sum_{n=2..m} -r(n)/(n*p^n), where r(n) = rootSum(1 - 2*x - x^2 + x^3, x^n) - 1.

Example

0.147349400002001458076808431847692596396671858173272158442...

Mathematica

digits = 66; m0 = 1000; dm = 100; Clear[s]; r[n_] := RootSum[1 - 2*# - #^2 + #^3& , #^n&] - 1; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> m0, WorkingPrecision -> 400] // Exp; s[m0]; s[m = m0 + dm]; While[RealDigits[s[m], 10, digits][[1]] != RealDigits[s[m - dm], 10, digits][[1]], Print[m]; m = m + dm]; RealDigits[s[m]][[1]]

Prog

(PARI) prodeulerrat(1 - (p^2 - (p - 1)^2)/(p^2*(p - 1))) \\ Amiram Eldar, Mar 16 2021

Crossrefs

Cf. A005596.

Sequence in context: A197697 A241026 A198743 * A190357 A296499 A199446

Adjacent sequences: A271795 A271796 A271797 * A271799 A271800 A271801

Keyword

nonn,cons

Author

Jean-François Alcover, Apr 14 2016

Extensions

More digits from Vaclav Kotesovec, Jun 19 2020

Status

approved