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).
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
Examples
0.35690496524995707612200530201399645913606668262573...
References
- V. I. Arnold, Dynamics, Statistics, and Projective Geometry of Galois Fields, Cambridge University Press, Cambridge, 2011, p. 78.
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.
- Marcin Mazur and Bogdan V. Petrenko, Representations of analytic functions as infinite products and their application to numerical computations, The Ramanujan Journal, Vol. 34, No. 1 (2014), pp. 129-141; arXiv preprint, arXiv:1202.1335 [math.NT], 2012.
Programs
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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]]
Formula
Equals (1/(2*sqrt(Pi))) * Product_{p prime} p^(3/2) * log(1 + 1/p) / sqrt(p-1).