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

0,1

Links

Table of n, a(n) for n=0..86.

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

Cf. A000005, A000203, A057020, A057021.

Sequence in context: A005623 A335110 A152989 * A261028 A195481 A199730

Adjacent sequences: A308048 A308049 A308050 * A308052 A308053 A308054

Keyword

nonn,cons

Author

Amiram Eldar, May 10 2019

Status

approved