A271869 Decimal expansion of Matthews' constant C_3, an analog of Artin's constant for primitive roots.

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

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_3 = Product_{p prime} 1 - (p^3 - (p - 1)^3)/(p^3*(p - 1)).

Example

0.0608216551203050860056322754619208554313373734757679419826434...

Mathematica

digits = 70; $MaxExtraPrecision = 1000; m0 = 2000; dm = 200; Clear[s]; LR =
LinearRecurrence[{2, 2, -6, 4, -1}, {0, 6, 0, 22, 5}, 2 m0]; r[n_Integer] := LR[[n]]; s[m_] := s[m] = NSum[-r[n] PrimeZetaP[n]/n, {n, 2, m}, NSumTerms -> 2 m0, WorkingPrecision -> digits+10] // 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]; Join[{0}, RealDigits[ s[m], 10, digits][[1]]]

Prog

(PARI) prodeulerrat(1 - (p^3 - (p - 1)^3)/(p^3*(p - 1))) \\ Amiram Eldar, Mar 16 2021

Crossrefs

Cf. A005596, A065414, A065415, A065416, A271780, A271798.

Sequence in context: A279929 A244812 A083680 * A010491 A257095 A159845

Adjacent sequences: A271866 A271867 A271868 * A271870 A271871 A271872

Keyword

nonn,cons

Author

Jean-François Alcover, Apr 16 2016

Extensions

More digits from Vaclav Kotesovec, Jun 19 2020

Status

approved