A308042 - OEIS

%I #14 Sep 16 2024 02:36:18

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

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

%U 0,8,6,0,5,3,8,4,4,8,9,9,3,9,9,3,3,5,7

%N Decimal expansion of the asymptotic mean of d_3(k)/ud_3(k), where d_3(k) is the number of ordered factorizations of k as product of 3 divisors (A007425) and ud_3(k) = 3^omega(k) is the unitary analog of d_3 (A074816).

%H Meselem Karras and Abdallah Derbal, <a href="https://doi.org/10.1142/S179355712050062X">Mean value of an arithmetic function associated with the Piltz divisor function</a>, Asian-European Journal of Mathematics, Vol. 13, No. 3 (2018), 2050062.

%F Equals Product_{p prime} ((1 - 1/p) * (2 + (1 - 1/p)^(-3))/3).

%e 2.22416248380186958442174889454690037857600080851428...

%t $MaxExtraPrecision = 1000; m = 1000; c = LinearRecurrence[{6, -16, 64/3, -32/3}, {0, 8, 32, 224/3}, m]; RealDigits[Exp[NSum[Indexed[c, n]*PrimeZetaP[n]/n/2^n, {n, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]

%o (PARI) prodeulerrat((1 - 1/p) * (2 + (1 - 1/p)^(-3))/3) \\ _Amiram Eldar_, Sep 16 2024

%Y Cf. A001221, A007425, A074816, A307869.

%K nonn,cons

%O 1,1

%A _Amiram Eldar_, May 10 2019