This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A113865 #18 Jun 22 2022 14:45:15 %S A113865 1,1,1,1,2,2,3,3,4,5,6,6,7,8,9,10,11,11,12,13,14,15,16,17,18,19,20,21, %T A113865 22,23,24,26,27,28,29,30,31,32,33,35,36,37,38,39,40,42,43,44,45,47,48, %U A113865 49,50,52,53,54,55,57,58,59,60,62,63,64,66,67,68,70,71,72,74,75,76,78 %N A113865 Number of digits of Bell number A000110(n). %C A113865 The positive integers which are in the complement to this sequence are: 25, 34, 41, 46, 51, 56, 61, 65, 69, 73, 77, 80, 84, 88, 91, 94, 98, 101, ... because there is no Bell number with 25 digits (B(30) = 846749014511809332450147 has 24 digits, B(31) = 10293358946226376485095653 has 26 digits). %C A113865 Since a(n) >> n log n, there are infinitely many numbers (indeed, almost all positive integers) in the complement of this sequence. [_Charles R Greathouse IV_, Aug 10 2011] %H A113865 Charles R Greathouse IV, <a href="/A113865/b113865.txt">Table of n, a(n) for n = 0..10000</a> %H A113865 John Sokol, <a href="http://www.dnull.com/bells/bell1000.html">The First 1000 Bell Numbers</a>. %F A113865 a(n) = ceiling(log_10 A000110(n)). %F A113865 a(n) ~ nk log n with k = 1/log 10. More specifically, a(n) = (n log n + n log log n - n + n/W(n) + log n - 0.5 log W(n) - 1)/log 10 + o(1), where W is Lambert's W function W(x)*exp(W(x)) = x. [_Charles R Greathouse IV_, Aug 11 2011] %e A113865 a(0) = 1 because Bell(0) = 1, which has one digit. %e A113865 a(1) = 1 because Bell(1) = 1, which has one digit. %e A113865 a(2) = 1 because Bell(2) = 2, which has one digit. %e A113865 a(3) = 1 because Bell(3) = 5, which has one digit. %e A113865 a(4) = 2 because Bell(4) = 15, which has two digits. %p A113865 seq(length(bell(n)), n = 0 .. 73); # _Zerinvary Lajos_, Aug 07 2007 %o A113865 (Python) %o A113865 from sympy import bell %o A113865 def A113865(n): return len(str(bell(n))) # _Chai Wah Wu_, Jun 22 2022 %Y A113865 A113015(n) = a(10^n). %Y A113865 Cf. A000110. %K A113865 base,nonn %O A113865 0,5 %A A113865 _Jonathan Vos Post_, Jan 25 2006