A138908 -id:A138908 - cp's OEIS frontend

A138902 a(n) = d!, where d is the number of digits in n.

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 6, 6, 6

Offset: 0

Odimar Fabeny, May 16 2008

James Spahlinger, Table of n, a(n) for n = 0..10000

Different from A047726. Cf. A138908.

a138902 = a000142 . a055642 -- James Spahlinger, Oct 09 2012

a[n_]:=IntegerLength[n]!;Array[a,103,0] (* James C. McMahon, Jun 24 2025 *)

a(n) = A000142(A055642(n)). - James Spahlinger, Oct 09 2012

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

A308314 Decimal expansion of Sum_{k>=1} (1/A055642(k)^A055642(k)) where A055642(k) is the number of digits of the integer k.

Original entry on oeis.org

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

Offset: 3

Bernard Schott, May 19 2019

With summation by parts to obtain 1st formula:
Sum_{k>=1} (1/length(k)^length(k)) =
Sum_{m=1..9} (1/1^1) + Sum_{m=10..99} (1/2^2) + Sum_{m=100...999} (1/3^3) + Sum_{m=1000...9999} (1/4^4) + ... =
9*(1/1^1) + 90*(1/2^2) + 900*(1/3^3) + 9000*(1/4^4) + 90000*(1/5^5) + ... =
9 ( 1/1^1 + 10^1/2^2 + 10^2/3^3 + 10^3/4^4 + 10^4/5^5 + ... =
(9/10) * (10^1/1^1 + 10^2/2^2 + 10^3/3^3 + 10^4/4^4 + 10^5/5^5 + ... =
(9/10) * ( (10/1)^1 + (10/2)^2 + (10/3)^3 + (10/4)^4 + (10/5)^5 + ... =
(9/10) * Sum_{m>=1} (10/m)^m.

			168.05245375262168949085673320556724...

Xavier Merlin, Methodix Analyse, Ellipses, 1997, Exercice 22 p. 120.
J.-M. Monier, Analyse, Tome 3, 2ème année, MP.PSI.PC.PT, Dunod, 1997, Exercice 3.2.1.h" p. 248.

Cf. A055642, A138908.

evalf((9/10) * Sum((10/n)^n, n=1..infinity), 100);

(9/10) * suminf(k=1, (10/k)^k) \\ Michel Marcus, Jun 08 2019

Equals (9/10) * Sum_{k>=1} (10/k)^k.

Equals Sum_{n>=1} (1/A138908(n)).

A196222 a(n) = A047726(n)^A047726(n), where A047726(n) is the number of numbers which can be written with n's digits.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 1, 27, 27, 46656

Offset: 0

M. F. Hasler, Sep 29 2011

Suggested by Franklin T. Adams-Watters and N. J. A. Sloane in analogy to A138908.

Cf. A138902, A138908.

cp's OEIS Frontend

A138902 a(n) = d!, where d is the number of digits in n.

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A308314 Decimal expansion of Sum_{k>=1} (1/A055642(k)^A055642(k)) where A055642(k) is the number of digits of the integer k.

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A196222 a(n) = A047726(n)^A047726(n), where A047726(n) is the number of numbers which can be written with n's digits.

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