A367110 Decimal expansion of Sum_{k has exactly 3 bits equal to 1 in base 2} 1/k.

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, 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, 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

Offset: 1

Author

Tengiz Gogoberidze, Dec 16 2023

Keywords

• cons
• nonn
• base

Comments

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

Examples

			1.4285915458526381...
		

Links

• Robert Baillie, Summing The Curious Series Of Kempner and Irwin, arXiv:0806.4410 [math.CA], 2008-2015.
• Wolfram Library Archive, KempnerSums.nb (8.6 KB) - Mathematica Notebook, Summing Kempner's Curious (Slowly-Convergent) Series

Crossrefs

Cf. A179951, A014311.

Programs

Mathematica

RealDigits[iSum[1, 3, 105, 2]][[1]] (* Amiram Eldar, Dec 16 2023, using Baillie's irwinSums.m *)

Formula

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

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

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