A308314 -id:A308314 - cp's OEIS frontend

A138908 a(n) = d^d, where d is the number of digits in n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27, 27

Offset: 0

Odimar Fabeny, May 16 2008

Cf. A055642, A138902, A308314.

dd[n_]:=Module[{d=IntegerLength[n]},d^d]; Join[{1},Array[dd,150]] (* Harvey P. Dale, Mar 16 2013 *)

a(n) = my(d=#Str(n)); d^d; \\ Michel Marcus, Nov 18 2020

From Amiram Eldar, Nov 18 2020: (Start)
a(n) = A055642(n)^A055642(n).
Sum_{n>=1} 1/a(n) = A308314. (End)

Edited by N. J. A. Sloane, Sep 29 2011, at the suggestion of Franklin T. Adams-Watters

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

Original entry on oeis.org

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

Bernard Schott, Sep 08 2020

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

			2.5584278811044952044644349496492935640012238762541921955925865566

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

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

Cf. A308314.

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

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

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

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

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

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

cp's OEIS Frontend

A138908 a(n) = d^d, where d is the number of digits in n.

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