A271877 Decimal expansion of Matthews' constant C_4, an analog of Artin's constant for primitive roots.

0, 2, 6, 1, 0, 7, 4, 4, 6, 3, 1, 4, 9, 1, 7, 7, 0, 8, 0, 8, 3, 2, 4, 9, 3, 9, 4, 3, 1, 3, 8, 2, 1, 4, 6, 7, 2, 5, 5, 6, 2, 6, 6, 7, 3, 6, 4, 0, 5, 5, 3, 8, 0, 4, 5, 2, 7, 6, 1, 1, 7, 3, 3, 7, 1, 0, 2, 4, 9, 8, 2, 0, 0, 5, 6, 5, 8, 7, 0, 1, 4, 0, 9, 9, 6, 8, 4, 7, 0, 4, 8, 1, 5, 1, 1, 5, 2, 2, 6, 0, 3, 8, 6, 9, 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_4 = Product_{p prime} 1 - (p^4 - (p - 1)^4)/(p^4*(p - 1)).

Example

0.026107446314917708083...

Mathematica

$MaxExtraPrecision = 2000; LR = LinearRecurrence[{2, 3, -10, 10, -5, 1}, {0, -8, 6, -40, 35, -194}, 10^4]; r[n_Integer] := LR[[n]]; NSum[r[n] PrimeZetaP[n]/n, {n, 2, Infinity}, NSumTerms -> 2000, WorkingPrecision -> 300, Method -> "AlternatingSigns"] // Exp // RealDigits[#, 10, 20]& // First // Prepend[#, 0]&
$MaxExtraPrecision = 1000; Clear[f]; f[p_] := 1 - (p^4 - (p - 1)^4)/(p^4*(p - 1)); Do[c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]]; Print[f[2] * Exp[N[Sum[Indexed[c, n]*(PrimeZetaP[n] - 1/2^n), {n, 2, m}], 105]]], {m, 100, 1000, 100}] (* Vaclav Kotesovec, Jun 19 2020 *)

Prog

(PARI) prodeulerrat(1 - (p^4 - (p - 1)^4)/(p^4*(p - 1))) \\ Amiram Eldar, Mar 16 2021

Crossrefs

Cf. A005596, A065414, A065415, A065416, A271780, A271798, A271869.

Sequence in context: A285849 A243398 A175474 * A021387 A321172 A127508

Adjacent sequences: A271874 A271875 A271876 * A271878 A271879 A271880

Keyword

nonn,cons

Author

Jean-François Alcover, Apr 16 2016

Extensions

More digits from Vaclav Kotesovec, Jun 19 2020

Status

approved