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Decimal expansion of Sum_{powerful k>=1} k / 2^k.
1

%I #6 Mar 12 2024 02:47:35

%S 7,9,9,0,7,3,2,1,9,8,2,3,2,7,2,3,9,9,4,4,1,0,6,4,3,2,3,2,1,0,0,5,0,6,

%T 4,5,3,1,7,2,2,2,4,6,5,3,9,8,5,2,9,3,3,3,2,9,0,2,0,9,1,2,5,8,6,8,5,9,

%U 4,6,2,8,9,9,4,7,8,6,2,6,1,1,7,8,0,3,9,2,2,4,6,6,8,9,2,2,3,7,4,3,8,5,7,8,7,1,8,5,5,0,5,9,6,1,8,8,6,2,6,2

%N Decimal expansion of Sum_{powerful k>=1} k / 2^k.

%C 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.

%F Equals Sum_{k>=1} A001694(k) / 2^A001694(k).

%F Equals Sum_{k>=1} k * A112526(k) / 2^k.

%e 0.79907321982327239944106432321005064531722246539852...

%t RealDigits[Sum[n/2^n, {n, Select[Range[1000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]}], 10, 120][[1]]

%Y Cf. A001694, A112526, A346173, A371134.

%K nonn,cons

%O 0,1

%A _Amiram Eldar_, Mar 12 2024