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%I #33 Jun 16 2021 14:05:36
%S 6,7,2,7,3,8,3,0,9,2,1,7,4,0,9,7,9,5,3,2,7,6,8,7,2,0,3,0,8,8,9,8,6,8,
%T 6,8,9,7,0,8,7,6,8,2,9,4,1,0,2,3,2,7,3,1,2,3,5,7,1,4,5,1,8,8,2,1,9,0,
%U 9,0,2,4,3,3,3,8,3,3,8,5,7,2,2,9,1,3,6,5,4,7,1,6,0,5,8,5,2,5,4,6,7,5,9,4,4,4
%N Decimal expansion of lim_{i->oo} c(i)/i, where c(i) is the number of integers k such that sigma(k) < i (A074753).
%C Erdös proved the existence of this constant. Dressler found its explicit form.
%H Vaclav Kotesovec, <a href="/A308039/b308039.txt">Table of n, a(n) for n = 0..1000</a>
%H Robert E. Dressler, <a href="https://doi.org/10.1016/0022-314X(72)90026-1">An elementary proof of a theorem of Erdös on the sum of divisors function</a>, Journal of Number Theory, Vol. 4, No. 6 (1972), pp. 532-536.
%H Paul Erdős, <a href="https://doi.org/10.1090/S0002-9904-1945-08390-6">Some remarks on Euler's phi function and some related problems</a>, Bulletin of the American Mathematical Society, Vol. 51, No. 8 (1945), pp. 540-544.
%H Leonid G. Fel, <a href="https://arxiv.org/abs/1108.0957">Summatory Multiplicative Arithmetic Functions: Scaling and Renormalization</a>, arXiv:1108.0957 [math.NT], 2011, p. 6.
%H Steven R. Finch, <a href="https://doi.org/10.1017/9781316997741">Mathematical Constants II</a>, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 51 (constant Y1).
%H V. Sita Ramaiah and D. Suryanarayana, <a href="http://ousar.lib.okayama-u.ac.jp/en/33820">Sums of reciprocals of some multiplicative functions</a>, Mathematical Journal of Okayama University, Vol. 21, No. 2 (1979), pp. 155-164, equation (4.2)-(4.3) on p. 162.
%H László Tóth, <a href="https://arxiv.org/abs/1608.00795">Alternating sums concerning multiplicative arithmetic functions</a>, arXiv:1608.00795 [math.NT], 2016.
%H László Tóth, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL20/Toth/toth25.html">Alternating sums concerning multiplicative arithmetic functions</a>, Journal of Integer Sequences, Vol. 20 (2017), Article 17.2.1 (section 4.3, p. 18).
%F Equals Product_{p prime} (1 - 1/p) * Sum_{k>=0} 1/sigma(p^k) = Product_{p prime} ((p - 1)^2/p) * Sum_{k>=1} 1/(p^k - 1) = Product_{p prime} 1 - ((p - 1)^2/p) * Sum_{k>=1} 1/((p^k - 1)*(p^(k+1) - 1)).
%F Equals lim_{n->oo} (1/log(n))*Sum_{k=1..n} 1/sigma(k).
%F Equals lim_{n->oo} (1/n) * Sum_{k=1..n} k/sigma(k) (the asymptotic mean of k/sigma(k)). - _Amiram Eldar_, Dec 23 2020
%F Sum_{k=1..n} 1/A000203(k) ~ Y1*log(n) + Y1*(gamma + Y2), where gamma = A001620, Y1 = A308039, Y2 = A345327. - _Vaclav Kotesovec_, Jun 14 2021
%e 0.6727383092174097953276872030889868689708768294102327312357145188219...
%t $MaxExtraPrecision = 1000; m = 1000; f[p_] := (1 - 1/p)*(p - 1)*Sum[1/(p^k - 1), {k, 1, m}]; c = Rest[CoefficientList[Series[Log[f[1/x]], {x, 0, m}], x]*Range[0, m]]; RealDigits[f[2]*Exp[NSum[Indexed[c, k]*(PrimeZetaP[k] - 1/2^k)/k, {k, 2, m}, NSumTerms -> m, WorkingPrecision -> m]], 10, 100][[1]]
%Y Cf. A000203, A074753, A345327.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, May 10 2019
%E More digits from _Vaclav Kotesovec_, Jun 13 2021
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