|
%I #24 Jan 14 2022 07:32:17
%S 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,
%T 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,
%U 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
%N Decimal expansion of Murata's constant Product_{p prime}(1 + 1/(p-1)^2).
%H Leo Murata, <a href="https://doi.org/10.1016/S0022-314X(05)80024-1">On the magnitude of the least prime primitive root</a>, Journal of Number Theory, Vol. 37, No. 1 (1991), pp. 47-66.
%H G. Niklasch, <a href="/A001692/a001692.html">Some number theoretical constants: 1000-digit values</a>. [Cached copy]
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/MuratasConstant.html">Murata's Constant</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeProducts.html">Prime Products</a>.
%F 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
%F 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
%e 2.8264199970675915755463917472369537490...
%t 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 *)
%o (PARI) prodeulerrat(1 + 1/(p-1)^2) \\ _Vaclav Kotesovec_, Sep 19 2020
%Y Cf. A000010, A000720, A008683, A078072, A241194, A241195.
%K cons,nonn
%O 1,1
%A _N. J. A. Sloane_, Nov 19 2001; edited Sep 16 2007 at the suggestion of _R. J. Mathar_
|
