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A327837
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Decimal expansion of the asymptotic mean of the number of exponential divisors function (A049419).
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10
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1, 6, 0, 2, 3, 1, 7, 1, 0, 2, 3, 0, 5, 4, 1, 8, 0, 5, 2, 3, 4, 9, 6, 2, 6, 3, 1, 5, 6, 2, 1, 1, 6, 1, 0, 0, 3, 7, 7, 6, 9, 3, 9, 4, 9, 5, 7, 8, 5, 5, 7, 2, 7, 3, 7, 7, 4, 6, 5, 3, 5, 2, 8, 5, 9, 8, 7, 8, 8, 8, 8, 6, 0, 2, 1, 6, 3, 3, 5, 4, 7, 2, 7, 5, 6, 6, 7, 3, 3, 9, 0, 4, 9, 4, 8, 8, 0, 6, 4, 1, 8, 0, 7, 5, 7
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OFFSET
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1,2
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LINKS
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Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 52 (constant Z3).
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FORMULA
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Equals Product_{p prime} (1 + Sum_{e >= 2} p^(-e) * (d(e) - d(e-1))), where d(e) is the number of divisors of e (A000005).
Equals Product_{p prime} (1 - 1/p) * (2 - (log(p-1) + QPolyGamma(0, 1, 1/p)) / log(p)). - Vaclav Kotesovec, Feb 27 2023
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EXAMPLE
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1.602317102305418052349626315621161003776939495785572...
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MATHEMATICA
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$MaxExtraPrecision = 1500; m = 1500; em = 500; f[x_] := 1 + Log[1 + Sum[x^e * (DivisorSigma[0, e] - DivisorSigma[0, e - 1]), {e, 2, em}]]; c = Rest[ CoefficientList[Series[f[x], {x, 0, m}], x] * Range[0, m] ]; RealDigits[ Exp[NSum[Indexed[c, k] * PrimeZetaP[k]/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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