A066022 Number of digits in n^n.

1, 1, 2, 3, 4, 5, 6, 8, 9, 11, 12, 13, 15, 17, 18, 20, 21, 23, 25, 27, 28, 30, 32, 34, 35, 37, 39, 41, 43, 45, 47, 49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 88, 90, 92, 94, 96, 98, 101, 103, 105, 107, 109, 112, 114, 116, 118, 121, 123, 125

Offset: 1

Author

Robert A. Stump (bee_ess107(AT)yahoo.com), Dec 11 2001

Keywords

• base
• easy
• nonn

Comments

This is almost certainly the same as the number of decimal digits of the sum of the n-th powers of the divisors of n (a sequence submitted by Labos Elemer on Jan 14 2002). Although no formal proof for this is known, Jon E. Schoenfield has verified it for n up to 10^8 and has given a plausible heuristic argument that it is true for all n.

Links

• Harry J. Smith, Table of n, a(n) for n=1..1000

Crossrefs

Cf. A000312, A054382, A055642.

Programs

Magma

[ #Intseq(n^n): n in [1..80] ]; // Vincenzo Librandi, Mar 08 2015

Maple

[seq(length(n^n), n=1..55)]; # Zerinvary Lajos, Mar 10 2007

Mathematica

Table[IntegerLength[n^n], {n, 80}] (* Vincenzo Librandi, Mar 08 2015 *)

PARI

a(n)=#digits(n^n);  \\ Joerg Arndt, Apr 30 2020

Formula

a(n) = A055642(A000312(n)). - Michel Marcus, Dec 05 2019

Extensions

Edited by N. J. A. Sloane Jan 03 2009 at the suggestion of Jon E. Schoenfield.