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

%I #49 Oct 19 2024 15:57:32

%S 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,

%T 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,

%U 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

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

%C This series is convergent.

%C 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.

%C 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

%H Jean-Paul Allouche, <a href="https://images-archive.math.cnrs.fr/Sommes-de-series-de-nombres-reels.html">Somme de séries de nombres réels</a>, Image des Mathématiques, CNRS, 2010 (in French).

%H Olivier Bordellès, Lixia Dai, Randell Heyman, Hao Pan, and Igor E. Shparlinski, <a href="https://arxiv.org/abs/1808.00188">On a sum involving the Euler function</a>, arXiv:1808.00188 [math.NT]

%H J. O. Shallit, <a href="https://www.jstor.org/stable/2322523">Solutions of Advanced Problems, 6450</a>, The American Mathematical Monthly, Vol. 92, No. 7, Aug.-Sep., 1985, pp. 513-514; DOI: 10.2307/2322523.

%H <a href="/index/Tra#transcendental">Index entries for transcendental numbers</a>

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

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

%e 2.5584278811044952044644349496492935640012238762541921955925865566

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

%t RealDigits[10*Log[10]/9, 10, 100][[1]] (* _Amiram Eldar_, Sep 08 2020 *)

%o (PARI) 10*log(10)/9 \\ _Charles R Greathouse IV_, Mar 22 2022

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

%Y Cf. A308314.

%K nonn,base,cons

%O 1,1

%A _Bernard Schott_, Sep 08 2020

%E a(90) corrected by _Georg Fischer_, Jul 12 2021