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A327574
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Decimal expansion of the constant that appears in the asymptotic formula for average order of the infinitary divisors sum function (A049417).
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2
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7, 3, 0, 7, 1, 8, 2, 4, 2, 1, 2, 7, 3, 8, 4, 2, 2, 5, 8, 3, 8, 9, 7, 5, 4, 6, 8, 1, 7, 3, 5, 3, 0, 1, 6, 1, 9, 5, 7, 2, 5, 6, 4, 3, 3, 8, 6, 1, 7, 2, 7, 8, 6, 9, 7, 0, 7, 3, 3, 6, 7, 6, 2, 3, 0, 1, 0, 7, 9, 8, 8, 3, 3, 2, 8, 0, 0, 5, 3, 4, 6, 3, 7, 0, 2, 9, 9
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OFFSET
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0,1
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COMMENTS
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The asymptotic mean of the infinitary abundancy index lim_{n->oo} (1/n) * Sum_{k=1..n} A049417(k)/k = 1.461436... is twice this constant. - Amiram Eldar, Jun 13 2020
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REFERENCES
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Steven R. Finch, Mathematical Constants II, Cambridge University Press, 2018, section 1.7.5, pp. 53-54.
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LINKS
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FORMULA
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Equals Limit_{k->oo} A327566(k)/k^2.
Equals (1/2) * Product_{P} (1 + 1/(P*(P+1))), where P are numbers of the form p^(2^k) where p is prime and k >= 0 (A050376).
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EXAMPLE
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0.730718242127384225838975468173530161957256433861727...
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MATHEMATICA
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$MaxExtraPrecision = 1000; m = 1000; em = 10; f[x_] := Sum[Log[1 + x^(2^e)/(1 + 1/x^(2^e))], {e, 0, em}]; c = Rest[CoefficientList[Series[f[x], {x, 0, m}], x]*Range[0, m]]; RealDigits[(1/2) * 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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STATUS
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approved
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