%I #17 Jun 16 2021 16:09:57
%S 5,0,7,3,3,8,8,8,2,5,8,3,0,8,4,3,7,8,1,0,0,4,9,7,8,7,6,5,1,5,9,5,2,6,
%T 7,7,3,8,9,0,1,9,6,3,4,8,2,8,1,6,4,4,8,0,8,0,4,9,7,4,5,8,7,7,2,4,5,0,
%U 6,9,4,6,1,7,3,0,2,8,6,5,1,6,3,0,0,5,6,8,8,3,9,1,7,6,3,0,2,4,6,5,9,6,0,5,8,0
%N Decimal expansion of a constant Y2 related to the asymptotics of A000203.
%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 Y2).
%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. 161-162, formula (4.1)-(4.4).
%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 Sum_{p primes} (p-1)^2 * g(p) * log(p) / (p*f(p)), where f(p) = 1 - (p-1)^2/p * Sum_{j>=1} 1/((p^j - 1)*(p^(j+1) - 1)) and g(p) = Sum_{j>=1} j/((p^j - 1)*(p^(j+1) - 1)).
%F Sum_{k=1..n} 1/A000203(k) ~ Y1*log(n) + Y1*(gamma + Y2), where gamma = A001620, Y1 = A308039, Y2 = A345327.
%e 0.5073388825830843781004978765159526773890196348281644808049...
%t $MaxExtraPrecision = 1000; Do[ratfun = (p - 1)^2 * Sum[j/(p^j - 1)/(p^(j + 1) - 1), {j, 1, m}]/(p*(1 - (p - 1)^2/p * Sum[1/(p^j - 1)/(p^(j + 1) - 1), {j, 1, m}])); zetas = 0; ratab = Table[konfun = Together[ratfun + c/(p^power - 1)]; coefs = CoefficientList[Numerator[konfun], p]; sol = Solve[Last[coefs] == 0, c][[1]]; zetas = zetas + c*Zeta'[power]/Zeta[power] /. sol; ratfun = konfun /. sol, {power, 2, 40}]; Print[N[Sum[Log[p]*ratfun /. p -> Prime[k], {k, 1, m}] + zetas, 110]], {m, 10, 250, 10}]
%Y Cf. A000203, A308039.
%K nonn,cons
%O 0,1
%A _Vaclav Kotesovec_, Jun 14 2021