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A308051
Decimal expansion of lim_{m->oo} (sqrt(log(m))/m^2) Sum_{k=1..m} sigma(k)/d(k), where d(k) is the number of divisors of k (A000005) and sigma(k) is their sum (A000203).
0
3, 5, 6, 9, 0, 4, 9, 6, 5, 2, 4, 9, 9, 5, 7, 0, 7, 6, 1, 2, 2, 0, 0, 5, 3, 0, 2, 0, 1, 3, 9, 9, 6, 4, 5, 9, 1, 3, 6, 0, 6, 6, 6, 8, 2, 6, 2, 5, 7, 3, 8, 4, 4, 2, 9, 6, 8, 7, 8, 8, 0, 2, 0, 1, 2, 7, 7, 4, 3, 4, 4, 2, 1, 4, 1, 8, 7, 2, 1, 3, 8, 5, 5, 3, 2, 1, 5
OFFSET
0,1
LINKS
Paul T. Bateman, Paul Erdös, Carl Pomerance, and E. G. Straus, The arithmetic mean of the divisors of an integer, in: Marvin I. Knopp (ed.), Analytic Number Theory, Proceedings of a Conference Held at Temple University, Philadelphia, May 12-15, 1980, Lecture Notes in Mathematics, Vol 899, Springer, Berlin, Heidelberg, 1981, pp. 197-220, alternative link.
Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 162.
FORMULA
Equals (1/(2*sqrt(Pi))) * Product_{p prime} p^(3/2) * log(1 + 1/p) / sqrt(p-1).
EXAMPLE
0.35690496524995707612200530201399645913606668262573...
MATHEMATICA
$MaxExtraPrecision = 1000; m = 1000; f[x_] := Log[1 + x]/x/Sqrt[1 - x]; c = Rest[CoefficientList[Series[Log[f[x]], {x, 0, m}], x]]; RealDigits[(1/2/ Sqrt[Pi])*Exp[NSum[Indexed[c, k]*PrimeZetaP[k], {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, May 10 2019
STATUS
approved