BellB[100]
In[]:=
47585391276764833658790768841387207826363669686825611466616334637559114497892442622672724044217756306953557882560751
Out[]=
BellB
DiscreteAsymptotic[BellB[n],{n,Infinity,1}]
In[]:=
-1-n+
n
ProductLog[n]

1
2
+n
n
ProductLog[n]
n
Out[]=
AsymptoticGreater[BellB[n],n^n,n->Infinity]
In[]:=
AsymptoticGreater[BellB[n],
n
n
,n∞]
Out[]=
AsymptoticGreater[BellB[n],Exp[n],n->Infinity]
In[]:=
AsymptoticGreater[BellB[n],
n

,n∞]
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/(n^n)],{n,100}]]]
In[]:=
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/(2^n)],{n,100}]]]
In[]:=
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/(3^n)],{n,100}]]]
In[]:=
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n])^n)],{n,100}]]]
In[]:=
Power
:Infinite expression
1
0
encountered.
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n]^2)^n)],{n,100}]]]
In[]:=
Power
:Infinite expression
1
0
encountered.
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n]^1.8)^n)],{n,100}]]]
In[]:=
Power
:Infinite expression
1
0.
encountered.
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n]^1.7)^n)],{n,100}]]]
In[]:=
Power
:Infinite expression
1
0.
encountered.
Out[]=
N[Sqrt[3]]
In[]:=
1.73205
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n]^Sqrt[3])^n)],{n,100}]]]
In[]:=
Power
:Infinite expression
1
0
encountered.
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n]^1.5)^n)],{n,100}]]]
In[]:=
Power
:Infinite expression
1
0.
encountered.
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n]^1.65)^n)],{n,100}]]]
In[]:=
Power
:Infinite expression
1
0.
encountered.
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n]^1.5)^n)],{n,100}]]]
ListLinePlot[Log10[Table[N[BellB[n]/(n!)],{n,100}]]]
In[]:=
Out[]=
Plot[Sum[Log10[N[BellB[n]/((Log[n]^a)^n)]],{n,10,100}],{a,1.5,1.8}]
In[]:=
Out[]=
FindRoot[Sum[Log10[N[BellB[n]/((Log[n]^a)^n)]],{n,10,100}],{a,1.5,1.8}]
In[]:=
{a1.65302}
Out[]=
FindRoot[Sum[Log10[N[BellB[n]/((Log[n]^a)^n)]],{n,10,200}],{a,1.5,1.8}]
In[]:=
{a1.8004}
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n]^2)^n)],{n,200}]]]
In[]:=
Power
:Infinite expression
1
0
encountered.
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n]^2)^n)],{n,200}]]]
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n])^n)],{n,200}]]]
In[]:=
Power
:Infinite expression
1
0
encountered.
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((Log[n])^(nLog[n]))],{n,200}]]]
In[]:=
Power
:Indeterminate expression
0
0
encountered.
General
:
1
466.127
4.61512
is too small to represent as a normalized machine number; precision may be lost.
General
:
1
471.747
4.62497
is too small to represent as a normalized machine number; precision may be lost.
General
:
1
477.377
4.63473
is too small to represent as a normalized machine number; precision may be lost.
General
:Further output of General::munfl will be suppressed during this calculation.
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((2)^(nLog[n]))],{n,200}]]]
In[]:=
General
:
1
1028.23
2.
is too small to represent as a normalized machine number; precision may be lost.
General
:
1
1034.51
2.
is too small to represent as a normalized machine number; precision may be lost.
General
:
1
1038.79
2.
is too small to represent as a normalized machine number; precision may be lost.
General
:Further output of General::munfl will be suppressed during this calculation.
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((2)^N[(nLog[n])])],{n,1000}]]]
In[]:=
Out[]=
ListLinePlot[Log10[Table[N[BellB[n]/((2)^N[(nLog[n])])],{n,10000}]]]
In[]:=
$Aborted
Out[]=
est[n_]:=N[BellB[n]/((2)^N[(nLog[n])])]
In[]:=
est[1000]
In[]:=
1.08172×
-152
10
Out[]=
est[2000]
In[]:=
7.84819×
-228
10
Out[]=
est[3000]
In[]:=
8.71715×
-267
10
Out[]=
est[4000]
In[]:=
4.51156×
-281
10
Out[]=
est[5000]
In[]:=
1.77368×
-276
10
Out[]=
est[6000]
In[]:=
1.57409×
-256
10
Out[]=
est[7000]
In[]:=
1.29238×
-223
10
Out[]=
est[8000]
In[]:=
1.83435×
-179
10
Out[]=
est[9000]
In[]:=
2.19564×
-125
10
Out[]=
est[10000]
In[]:=
2.06428×
-62
10
Out[]=
est[10500]
In[]:=
7.42663×
-28
10
Out[]=
est[10700]
In[]:=
1.72452×
-13
10
Out[]=
est[10800]
In[]:=
3.41454×
-6
10
Out[]=
est[10850]
In[]:=
0.016211
Out[]=
est[10852]
In[]:=
0.0227644
Out[]=
est[10860]
In[]:=
0.0885819
Out[]=
est[10870]
In[]:=
0.484872
Out[]=
est[10874]
In[]:=
0.957522
Out[]=
est[10875]
In[]:=
1.13514
Out[]=
est[10888]
In[]:=
10.3851
Out[]=
est[10890]
In[]:=
14.6025
Out[]=
est[10900]
In[]:=
80.3417
Out[]=
est[11000]
In[]:=
2.24313×
9
10
Out[]=
Table[est[n],{n,20}]
In[]:=
{1.,0.765092,0.509125,0.321238,0.196588,0.11783,0.0695918,0.0406636,0.0235708,0.0135798,0.00778704,0.00444914,0.0025349,0.00144115,0.000817983,0.000463713,0.000262647,0.000148675,0.0000841295,0.0000475979}
Out[]=
Table[BellB[n]/2^n,{n,20}]
In[]:=

1
2
,
1
2
,
5
8
,
15
16
,
13
8
,
203
64
,
877
128
,
1035
64
,
21147
512
,
115975
1024
,
339285
1024
,
4213597
4096
,
27644437
8192
,
95449661
8192
,
1382958545
32768
,
10480142147
65536
,
20716217451
32768
,
682076806159
262144
,
5832742205057
524288
,
12931039558843
262144

Out[]=
N[%]
In[]:=
{0.5,0.5,0.625,0.9375,1.625,3.17188,6.85156,16.1719,41.3027,113.257,331.333,1028.71,3374.57,11651.6,42204.5,159914.,632209.,2.60192×
6
10
,1.11251×
7
10
,4.9328×
7
10
}
Out[]=
est[10000]
Table[2^n<BellB[n],{n,10}]
In[]:=
{False,False,False,False,True,True,True,True,True,True}
Out[]=