A334388 Decimal expansion of Sum_{k>=1} A007953(k) / (k*(k+1)) where A007953(k) is the sum of digits of the integer k.

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

Offset: 1

Author

Bernard Schott, Sep 08 2020

Keywords

• nonn
• base
• cons

Comments

This series is convergent.

Jeffrey Shallit generalizes this result to any base b (see Amer. Math. Month. link): Sum_{k>=1} digsum(k)_b / (k*(k+1)) = (b/(b-1)) * log(b) where digsum(k)_b is the sum of the digits of k when expressed in base b.

Sum_{n <= x} s(floor(x/n)) = kx + O(x^(2/3 + o(1))) where s(n) is the digital sum A007953 and k is this constant. See Bordellès, Dai, Heyman, Pan, & Shparlinski, Example 3.4. - Charles R Greathouse IV, Mar 22 2022

Examples

			2.5584278811044952044644349496492935640012238762541921955925865566
		

Links

• Jean-Paul Allouche, Somme de séries de nombres réels, Image des Mathématiques, CNRS, 2010 (in French).
• Olivier Bordellès, Lixia Dai, Randell Heyman, Hao Pan, and Igor E. Shparlinski, On a sum involving the Euler function, arXiv:1808.00188 [math.NT]
• J. O. Shallit, Solutions of Advanced Problems, 6450, The American Mathematical Monthly, Vol. 92, No. 7, Aug.-Sep., 1985, pp. 513-514; DOI: 10.2307/2322523.
• Index entries for transcendental numbers

Crossrefs

Cf. A002392 (log(10)), A007953 (digsum), A016627 (for base 2).

Cf. A308314.

Programs

Maple

evalf(10*log(10)/9,90);

Mathematica

RealDigits[10*Log[10]/9, 10, 100][[1]] (* Amiram Eldar, Sep 08 2020 *)

PARI

10*log(10)/9 \\ Charles R Greathouse IV, Mar 22 2022

Formula

Equals 1/(1*2) + 2/(2*3) + 3/(3*4) + 4/(4*5) + ... + 1/(10*11) + 2/(11*12) + ...

Equals (10/9) * log(10).

Extensions

a(90) corrected by Georg Fischer, Jul 12 2021