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%I #14 Mar 12 2024 09:25:18
%S 1,6,9,7,9,0,7,8,1,9,7,7,9,6,2,5,0,6,4,4,6,4,2,4,0,8,9,9,6,5,3,4,7,8,
%T 9,1,8,4,3,6,3,5,1,5,3,1,8,8,6,2,4,7,2,6,3,4,0,6,9,9,8,6,0,8,9,0,8,9,
%U 5,4,1,2,9,0,6,1,4,3,9,7,7,3,9,2,0,3,0,0,8,6,5,3,4,4,7,1,8,7,7,5,2,9,5,0,4
%N Decimal expansion of Sum_{squarefree k>=1} k / 2^k.
%C Erdős (1981) conjectured and Chen and Ruzsa (1999) proved that this constant is irrational.
%H Yong-Gao Chen and Imre Z. Ruzsa, <a href="https://doi.org/10.1023/A:1004742930674">On the irrationality of certain series</a>, Periodica Mathematica Hungarica, Vol. 38, No. 1 (1999), pp. 31-37.
%H Paul Erdős, <a href="https://users.renyi.hu/~p_erdos/1981-36.pdf">Sur l'irrationalité d'une certaine série</a>, C. R. Acad. Sci. Paris, Sér. 1, Vol. 292 (1981), pp. 765-768.
%F Equals Sum_{k>=1} A005117(k) / 2^A005117(k).
%F Equals Sum_{k>=1} k * mu(k)^2 / 2^k.
%e 1.69790781977962506446424089965347807016709423133847...
%t RealDigits[Sum[n/2^n, {n, Select[Range[1000], SquareFreeQ]}], 10, 120][[1]]
%Y Cf. A005117, A008683, A333182, A346173, A371135.
%K nonn,cons
%O 1,2
%A _Amiram Eldar_, Mar 12 2024
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