A271780 Decimal expansion of Product_{p odd prime} 1-2/(p*(p-1)), a constant related to Artin's conjecture in the context of quadratic fields.

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

Offset: 0

Author

Jean-François Alcover, Apr 15 2016

Keywords

• nonn
• cons

Examples

			0.5351070126166387332839586518606321598586393391028...
		

References

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

Crossrefs

Cf. A005597, A217739 (8/Pi^2).

Programs

Mathematica

digits = 99; $MaxExtraPrecision = 600; m0 = 1000; dm = 100; Clear[s]; LR = LinearRecurrence[{2, 1, -2}, {0, 4, 6}, 2 m0]; r[n_Integer] := LR[[n]];
s[m_] := s[m] = NSum[-r[n] (PrimeZetaP[n] - 1/2^n)/n, {n, 2, m}, NSumTerms -> m0, WorkingPrecision -> 600] // 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], 10, digits][[1]] (* Jean-François Alcover, Apr 15 2016 *)

PARI

prodeulerrat(1-2/(p*(p-1)), 1, 3) \\ Amiram Eldar, Mar 11 2021

Formula

Equals (8/Pi^2)*A005597.