A113015 -id:A113015 - cp's OEIS frontend

A113883 Indices of semiprime Bell numbers A000110.

4, 6, 16, 31, 33, 49, 84

Offset: 1

Jonathan Vos Post, Jan 27 2006

Semiprime analog of A051130 (indices of prime Bell numbers). These indices of semiprime Bell numbers include all values through B(120), which has 146 digits and at least 5 prime factors, the smallest being 71.

			a(1) = 4 because B(4) = 15 = 3 * 5.
a(2) = 6 because B(6) = 203 = 7 * 29.
a(3) = 16 because B(16) = 10480142147 = 241 * 43486067.
a(4) = 31 because B(31) = 10293358946226376485095653
= 11 * 935759904202397862281423.
a(5) = 33 because B(33) = 1629595892846007606764728147
= 5694673 * 286161451736738458339.
a(6) = 49 because B(49) =
10726137154573358400342215518590002633917247281
= 7615441337805454611187 *
1408472165798904899327563.
a(7) = 84 because B(84) is a 93-digit semiprime, whose smaller prime factor is 8429925224798761223.

John Sokol, The First 1000 Bell Numbers.

Cf. A000110, A001358, A113015.

n such that A000110(n) is semiprime. n such that A000110(n) is in A001358.

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

Original entry on oeis.org

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

Jonathan Vos Post, Jan 25 2006

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]

			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.

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

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

Cf. A000110.

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

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

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]

A113908 Number of prime factors, with multiplicity, of Bell number A000110(n).

Original entry on oeis.org

0, 0, 1, 1, 2, 3, 2, 1, 6, 4, 3, 4, 3, 1, 3, 3, 2, 7, 3, 4, 6, 4, 6, 4, 3, 6, 5, 6, 4, 6, 6, 2, 5, 2, 4, 7, 4, 3, 4, 3, 3, 6, 1, 7, 6, 5, 4, 8, 4, 2, 5, 3, 5, 6, 3, 1, 12, 3, 3, 5, 3, 7, 3, 7, 4, 5, 6, 3, 5, 4, 4, 10, 9, 6, 6, 5, 8, 5, 5, 8, 5, 4, 5, 3, 2

Offset: 0

Jonathan Vos Post, Jan 29 2006

nonn

This is 1 for A051330 (indices of prime Bell numbers) and is 2 for A113883 (indices of semiprime Bell numbers). The records begin a(0) = 0, a(2) = 1, a(4) = 2, a(5) = 3, a(8) = 6, a(17) = 7, a(56) = 12.

			a(5) = BigOmega(Bell(5)) = A001222(52) = A001222(2^2 * 13) = 3.

Amiram Eldar, Table of n, a(n) for n = 0..104
John Sokol, The First 1000 Bell Numbers.

Cf. A000110, A001222, A113015, A113883.

with(numtheory):with(combinat):a:=proc(n) if n=0 then 0 else bigomega(bell(n)) fi end: seq(a(n), n=0..43); # Zerinvary Lajos, Apr 11 2008

Table[PrimeOmega[BellB[n]], {n, 0, 50}] (* Amiram Eldar, Nov 23 2019 *)

a(n) = BigOmega(A000110(n)). a(n) = A001222(A000110(n)).

cp's OEIS Frontend

A113883 Indices of semiprime Bell numbers A000110.

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

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A113908 Number of prime factors, with multiplicity, of Bell number A000110(n).

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