A065485 Decimal expansion of Murata's constant Product_{p prime} (1 + 1/(p-1)^2).

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

Offset: 1

Author

N. J. A. Sloane, Nov 19 2001; edited Sep 16 2007 at the suggestion of R. J. Mathar

Keywords

• cons
• nonn

Examples

			2.8264199970675915755463917472369537490...
		

References

• Steven R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, Sections 2.4 and 2.7, pp. 106, 117.

Links

• Leo Murata, On the magnitude of the least prime primitive root, Journal of Number Theory, Vol. 37, No. 1 (1991), pp. 47-66.
• G. Niklasch, Some number theoretical constants: 1000-digit values. [Cached copy]
• Eric Weisstein's World of Mathematics, Murata's Constant.
• Eric Weisstein's World of Mathematics, Prime Products.

Crossrefs

Cf. A000010, A000720, A008683, A078072, A241194, A241195.

Programs

Mathematica

digits = 99; terms = 1000; $MaxExtraPrecision = 500; r[n_Integer] := 2 - (1-I)^(n+1) - (1+I)^(n+1); NSum[r[n-1]*PrimeZetaP[n]/n, {n, 2, terms}, NSumTerms -> terms, WorkingPrecision -> digits+10] // Exp // RealDigits[ #, 10, digits]& // First (* Jean-François Alcover, Apr 16 2016 *)

PARI

prodeulerrat(1 + 1/(p-1)^2) \\ Vaclav Kotesovec, Sep 19 2020

Formula

Equals lim_{k->oo} (1/pi(k)) * Sum_{p prime, p <= k} (p-1)/phi(p-1), where pi(k) = A000720(k) and phi(k) = A000010(k) (Murata, 1991). - Amiram Eldar, Jul 31 2020

Equals Sum_{k>=1} mu(k)^2/phi(k)^2, where mu is the Möbius function (A008683) and phi is the Euler totient function (A000010). - Amiram Eldar, Jan 14 2022