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Decimal expansion of Sum_{k has exactly 3 bits equal to 1 in base 2} 1/k.
1

%I #32 Dec 21 2023 11:19:19

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

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

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

%N Decimal expansion of Sum_{k has exactly 3 bits equal to 1 in base 2} 1/k.

%C For 1 bit equal to 1 the sum is 2, for 2 bits equal to 1 the sum is 1.52899956069688841838263949451... (see A179951).

%H Robert Baillie, <a href="https://arxiv.org/abs/0806.4410">Summing The Curious Series Of Kempner and Irwin</a>, arXiv:0806.4410 [math.CA], 2008-2015.

%H Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, <a href="https://library.wolfram.com/infocenter/MathSource/7166/">Summing Kempner's Curious (Slowly-Convergent) Series</a>

%F Equals Sum_{m>=2} Sum_{j=1..m-1} Sum_{i=0..j-1} 1/(2^i + 2^j + 2^m).

%F Equals 2 * Sum_{j>=2} Sum_{i=1..j-1} 1/(2^i + 2^j + 1).

%F Equals Sum_{k>=1} 1/A014311(k).

%e 1.4285915458526381...

%t RealDigits[iSum[1, 3, 105, 2]][[1]] (* _Amiram Eldar_, Dec 16 2023, using Baillie's irwinSums.m *)

%Y Cf. A179951, A014311.

%K cons,nonn,base

%O 1,2

%A _Tengiz Gogoberidze_, Dec 16 2023