A308042 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).
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, 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, 0, 8, 6, 0, 5, 3, 8, 4, 4, 8, 9, 9, 3, 9, 9, 3, 3, 5, 7
Offset: 1
Examples
2.22416248380186958442174889454690037857600080851428...
Links
- Meselem Karras and Abdallah Derbal, Mean value of an arithmetic function associated with the Piltz divisor function, Asian-European Journal of Mathematics, Vol. 13, No. 3 (2018), 2050062.
Programs
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Mathematica
$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]] -
PARI
prodeulerrat((1 - 1/p) * (2 + (1 - 1/p)^(-3))/3) \\ Amiram Eldar, Sep 16 2024
Formula
Equals Product_{p prime} ((1 - 1/p) * (2 + (1 - 1/p)^(-3))/3).