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

1,1

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

Table of n, a(n) for n=1..99.

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.

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

Example

2.8264199970675915755463917472369537490...

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 *)

Prog

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

Crossrefs

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

Sequence in context: A203022 A224196 A141449 * A229939 A237889 A064912

Adjacent sequences: A065482 A065483 A065484 * A065486 A065487 A065488

Keyword

cons,nonn,changed

Author

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

Status

approved