n! = a^b

[Wouter Meeussen]

hi,

would it make sense to define 2 sequences on the basis of:

a[n] is the best approximation to (n-1)! of the form a[n]^b with a[n]<=n 
b[n] is the best approximation to (n-1)! of the form a^b[n] with a   <=n

with a[n]^b[n] implied, since 
Log[ (n-1)!]/Log[a[n]]=~     b[n]   and
Log[ (n-1)!]/    b[n] =~ Log[a[n]]

of course, the relative error is soon les than a percent, but the absolute error
gets as humongous as n! does.

*****
I really don't know if 'best approximations' have a place in EIS.
do you think they belong there?
*****

a variation is the analogous sequence, where we allow the base b to be
*twice* n:
c[n] is the best approximation to (n-1)! of the form c[n]^d with c[n]<=2n
d[n] is the best approximation to (n-1)! of the form c^d[n] with c   <=2n
(it gives a better fit, at the detriment of 'pertinence').

a[n]  =
{2, 2, 4, 5, 5, 3, 8, 6, 5, 4, 7, 12, 5, 7, 16, 9, 11, 16, 6, 12, 17, 4, 3, 
21, 25, 25, 12, 4, 14, 9, 18, 23, 30, 5, 24, 8,  
  12, 25, 28, 11, 26, 19, 18, 27, 40, 18, 5, 9, 34, 45, 32, 26, 50, 17, 55, 
55, 12, 43, 6, 47, 38, 16, 32, 16, 56, 16, 30,  
  61, 66, 66, 37, 29, 28, 26, 15, 6, 24, 67, 40, 26, 37, 60, 68, 5, 49, 52, 
43, 19, 53, 83, 66, 16, 94, 94, 79, 10, 51, 89, 
  24, 25, 69, 100, 58, 77, 54, 33, 57, 86, 33, 39, 106, 15, 96, 101, 45, 77, 
112, 112, 69, 93, 107, 10, 98, 94, 118, 54, 11, 
  113, 124, 124, 48, 62, 74, 130, 30, 136, 35, 43, 106, 74, 38, 111, 107, 50, 
63, 16, 126, 17, 117, 105, 60, 35, 132, 55, 82, 
  123, 88, 128, 6, 108, 59, 161, 161, 161, 139, 37, 130, 44, 78, 58, 115, 119,
 54, 156, 103, 128, 110, 162, 114, 142, 168,  
  138, 134, 69, 71, 139, 120, 35, 37, 121, 103, 109, 193, 193, 15, 130, 100, 
160, 151}

b[n]=
{0, 1, 1, 2, 3, 6, 4, 6, 8, 11, 9, 8, 14, 13, 10, 14, 14, 13, 22, 17, 16, 35, 
47, 18, 18, 19, 26, 49, 27, 34, 27, 26, 25, 55,  
  29, 46, 40, 32, 32, 46, 35, 40, 42, 38, 35, 46, 85, 64, 41, 39, 44, 48, 41, 
58, 42, 43, 71, 48, 103, 49, 53, 71, 58, 74,  
  52, 77, 64, 54, 54, 55, 65, 71, 73, 76, 93, 143, 82, 63, 73, 84, 77, 69, 68,
 181, 76, 76, 81, 105, 79, 72, 77, 118, 73, 74,  
  78, 150, 89, 79, 113, 113, 87, 81, 93, 88, 97, 112, 98, 90, 116, 112, 89, 
155, 93, 93, 114, 101, 94, 95, 107, 101, 99, 203,  
  103, 105, 101, 122, 205, 105, 104, 105, 132, 125, 121, 108, 156, 109, 152, 
145, 118, 129, 154, 120, 122, 147, 140, 211,  
  122, 210, 126, 130, 149, 173, 127, 156, 143, 132, 143, 133, 363, 140, 162, 
131, 132, 133, 138, 190, 142, 184, 161, 174,  
  150, 150, 181, 144, 158, 152, 158, 147, 159, 153, 149, 156, 158, 184, 184, 
160, 166, 225, 223, 169, 176, 175, 157, 158,  
  309, 173, 184, 168, 171}

and 

c[n]=
 {2, 2, 6, 5, 11, 9, 16, 14, 13, 21, 7, 12, 25, 23, 22, 30, 21, 38, 36, 34, 
44, 32, 40, 50, 48, 46, 36, 54, 14, 35, 61, 59,  
   30, 56, 55, 46, 75, 73, 28, 11, 26, 19, 66, 57, 74, 18, 5, 71, 34, 90, 69, 
77, 61, 96, 55, 83, 92, 91, 81, 47, 66, 16, 32,  
   16, 86, 94, 85, 61, 123, 66, 37, 119, 77, 98, 127, 6, 150, 67, 40, 112, 
103, 142, 95, 5, 49, 149, 126, 19, 53, 83, 115,  
   166, 107, 114, 79, 100, 51, 113, 24, 106, 69, 127, 58, 188, 112, 33, 197, 
160, 194, 149, 132, 15, 96, 101, 45, 225, 124,  
   163, 194, 93, 107, 137, 190, 94, 130, 151, 11, 208, 207, 124, 205, 62, 157,
 253, 30, 164, 35, 261, 106, 163, 38, 229, 107,  
   252, 195, 16, 194, 247, 117, 105, 232, 255, 132, 219, 82, 263, 238, 161, 
216, 108, 59, 245, 214, 174, 139, 319, 231, 44,  
   78, 220, 297, 238, 54, 156, 187, 256, 110, 162, 114, 225, 325, 138, 134, 
69, 271, 139, 200, 249, 328, 248, 267, 109, 193,  
   193, 229, 237, 100, 273, 151},

d[n]=
 {0, 1, 1, 2, 2, 3, 3, 4, 5, 5, 9, 8, 7, 8, 
9, 9, 11, 10, 11, 12, 12, 14, 14, 14, 15, 16,  
   18, 17, 27, 21, 19, 20, 25, 22, 23, 25, 23, 24, 32, 46, 35, 40, 29, 31, 30,
 46, 85, 33, 41, 33, 36, 36, 39, 36, 42, 39,  
   39, 40, 42, 49, 46, 71, 58, 74, 47, 47, 49, 54, 47, 55, 65, 50, 56, 54, 52,
 143, 52, 63, 73, 58, 60, 57, 63, 181, 76, 60,  
   63, 105, 79, 72, 68, 64, 71, 71, 78, 75, 89, 75, 113, 78, 87, 77, 93, 73, 
82, 112, 75, 79, 77, 82, 85, 155, 93, 93, 114,  
   81, 92, 88, 86, 101, 99, 95, 90, 105, 99, 97, 205, 93, 94, 105, 96, 125, 
103, 95, 156, 105, 152, 98, 118, 109, 154, 104,  
   122, 104, 110, 211, 112, 108, 126, 130, 112, 111, 127, 116, 143, 114, 117, 
127, 121, 140, 162, 121, 125, 131, 138, 119,  
   127, 184, 161, 131, 125, 131, 181, 144, 140, 133, 158, 147, 159, 140, 132, 
156, 158, 184, 140, 160, 150, 145, 139, 147,  
   146, 175, 157, 158, 154, 154, 184, 152, 171}

wouter.

NV Vandemoortele Coordination Center
Oils & Fats Applied Research
Prins Albertlaan 79
Postbus 40
B-8870 Izegem (Belgium)
Tel: +/32/51/33 21 11
Fax: +/32/51/33 21 75
vdmcc at vandemoortele.be