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A247890 Number of digits in (R_n)^n.

%I A247890 #63 Nov 11 2024 01:57:11
%S A247890 1,3,7,13,21,31,43,57,73,91,111,133,157,183,211,241,273,307,343,381,
%T A247890 421,464,508,554,602,652,704,758,814,872,932,994,1058,1124,1192,1262,
%U A247890 1334,1408,1484,1562,1642,1724,1808,1895,1983,2073,2165,2259,2355,2453,2553,2655,2759,2865
%N A247890 Number of digits in (R_n)^n.
%C A247890 R_n is the n-th repunit (i.e., R_n = 11...111 with n 1's).
%C A247890 From _David A. Corneth_, Jun 27 2016: (Start)
%C A247890 The number of digits of m is floor(log(m)/log(10)) + 1 for m > 0.
%C A247890 R_n = (10^n - 1) / 9 = (10 - 10^(1-n))/9 * 10^(n-1). Its number of digits is floor(n * log((10 - 10^(1-n))/9) / log(10)) + n * (n - 1) + 1. [corrected by _Jason Yuen_, Nov 11 2024] (End)
%H A247890 Vincenzo Librandi, <a href="/A247890/b247890.txt">Table of n, a(n) for n = 1..1000</a>
%F A247890 a(n) = n^2 - n + 1 = A002061(n), for 1 <= n <= 21.
%F A247890 a(n) = n^2 - n + 2 = A014206(n-1), for 22 <= n <= 43.
%F A247890 a(n) = A055642(A245593(n)). - _Michel Marcus_, Apr 20 2022
%t A247890 Table[IntegerLength[((10^n - 1)/9)^n], {n, 54}] (* or *)
%t A247890 Table[IntegerLength[FromDigits[Table[1, {n}]]^n], {n, 54}] (* _Michael De Vlieger_, Jun 27 2016 *)
%o A247890 (PARI) vector(100,n,#Str(((10^n-1)/9)^n))
%o A247890 (PARI) a(n) = logint(((10 - 10^(1-n))/9)^n\1,10)+n^2-n+1 \\ _David A. Corneth_, Jun 27 2016
%o A247890 (Magma) [#Intseq(Floor((10^n-1)/9)^n): n in [1..50]]; // _Marius A. Burtea_, May 20 2019
%o A247890 (Python)
%o A247890 def a(n): return len(str(int("1"*n)**n))
%o A247890 print([a(n) for n in range(1, 55)]) # _Michael S. Branicky_, Apr 20 2022
%Y A247890 Cf. A002061, A014206, A002275.
%Y A247890 Cf. A055642, A245593.
%K A247890 nonn,easy,base
%O A247890 1,2
%A A247890 _Derek Orr_, Sep 25 2014
%E A247890 Incorrect conjectures removed by _Georg Fischer_, May 19 2019