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A113865 Number of digits of Bell number A000110(n). 1
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 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,5
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]
LINKS
Charles R Greathouse IV, Table of n, a(n) for n = 0..10000
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]
EXAMPLE
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.
MAPLE
seq(length(bell(n)), n = 0 .. 73); # Zerinvary Lajos, Aug 07 2007
PROG
(Python)
from sympy import bell
def A113865(n): return len(str(bell(n))) # Chai Wah Wu, Jun 22 2022
CROSSREFS
A113015(n) = a(10^n).
Cf. A000110.
Sequence in context: A285902 A097330 A086861 * A358636 A347609 A143738
KEYWORD
base,nonn
AUTHOR
Jonathan Vos Post, Jan 25 2006
STATUS
approved

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Last modified April 17 17:01 EDT 2024. Contains 371765 sequences. (Running on oeis4.)