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

1,1

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

Links

Table of n, a(n) for n=1..100.

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

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).

Example

2.5584278811044952044644349496492935640012238762541921955925865566

Maple

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

Mathematica

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

Prog

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

Crossrefs

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

Cf. A308314.

Sequence in context: A194531 A092394 A027438 * A204237 A153162 A168199

Adjacent sequences: A334385 A334386 A334387 * A334389 A334390 A334391

Keyword

nonn,base,cons

Author

Bernard Schott, Sep 08 2020

Extensions

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

Status

approved