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Decimal expansion of the (finite) value of Sum_{ k >= 1, k has only a single zero digit in base 2 } 1/k.
4

%I #13 Jul 14 2020 10:48:19

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

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

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

%N Decimal expansion of the (finite) value of Sum_{ k >= 1, k has only a single zero digit in base 2 } 1/k.

%C The sum of 1/n where n has a single 0 in base 2.

%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.

%F Equals Sum_{n>=2} Sum_{k=0..n-2}, 1/(2^n - 1 - 2^k).

%e 1.4625907350443646995461454467205346210747448647488211093642006243545229...

%t RealDigits[ N[ Sum[1/(2^n - 1 - 2^k), {n, 2, 400}, {k, 0, n - 2}], 111]][[1]]

%t (* first install irwinSums.m, see reference, then *) First@ RealDigits@ iSum[0, 1, 111, 2] (* _Robert G. Wilson v_, Aug 03 2010 *)

%Y Cf. A030130 (numbers with a single zero in base 2), A140502.

%K base,cons,nonn

%O 1,2

%A _Robert G. Wilson v_, May 15 2009