A113865 Number of digits of Bell number A000110(n).

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, 22, 23, 24, 26, 27, 28, 29, 30, 31, 32, 33, 35, 36, 37, 38, 39, 40, 42, 43, 44, 45, 47, 48, 49, 50, 52, 53, 54, 55, 57, 58, 59, 60, 62, 63, 64, 66, 67, 68, 70, 71, 72, 74, 75, 76, 78

Offset: 0

Author

Jonathan Vos Post, Jan 25 2006

Keywords

• base
• nonn

Comments

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).

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]

Examples

			a(0) = 1 because Bell(0) = 1, which has one digit.
a(1) = 1 because Bell(1) = 1, which has one digit.
a(2) = 1 because Bell(2) = 2, which has one digit.
a(3) = 1 because Bell(3) = 5, which has one digit.
a(4) = 2 because Bell(4) = 15, which has two digits.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
• John Sokol, The First 1000 Bell Numbers.

Crossrefs

A113015(n) = a(10^n).

Cf. A000110.

Programs

Maple

seq(length(bell(n)), n = 0 .. 73); # Zerinvary Lajos, Aug 07 2007

Python

from sympy import bell
def A113865(n): return len(str(bell(n))) # Chai Wah Wu, Jun 22 2022

Formula

a(n) = ceiling(log_10 A000110(n)).

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]