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2.Minimal numbers with persisitence p

For b>=p! :
min(p,b)=(p+1)b-[b/p!]
A proof is shown in [4].

This yields the first bound for pmax(b):
pmax(x)>sqrt(ln b)

For a divides b :
p_max(b)>=[(ln b + ln(a-1) / (ln a))]
A proof is also shown in [4].

As the first formula is defined only for bigger values of n, the samller ones must be examined differently. This is done at this place for all p<10. For p=10 more than 3.6 million min(p,b)-values needed to be determined.
For p<5 the values are shown in the following tables. For p>=5 please follow the a bit deeper standing links.

All values in decimal representation
b\p 0 1 2 3 4 5
2 0 2 - - - -
3 0 3 8 26 ? ?
4 0 4 10 63 ? ?
5 0 5 13 68 2344 244140624
6 0 6 15 23 172 3629
7 0 7 18 27 131 1601
8 0 8 20 31 174 1535
9 0 9 23 35 52 394
10 0 10 25 39 77 679
11 0 11 28 43 75 317
12 0 12 30 46 83 1099
13 0 13 33 50 75 127
14 0 14 35 54 81 135
15 0 15 38 58 89 582
16 0 16 40 62 95 187
17 0 17 43 66 101 168
18 0 18 45 69 104 157
19 0 19 48 73 110 201
20 0 20 50 77 133 159
21 0 21 53 81 143 159
22 0 22 55 85 127 230
23 0 23 58 89 133 215
24 0 24 60 92 119 180
All values in basis-representation
b\p 0 1 2 3 4 5
2 0 10 - - - -
3 0 10 22 222 ? ?
4 0 10 22 233 ? ?
5 0 10 23 233 33334 444444444444
6 0 10 23 35 444 24445
7 0 10 24 36 245 4445
8 0 10 24 37 256 2777
9 0 10 25 38 57 477
10 0 10 25 39 77 679
11 0 10 26 3'10 69 269
12 0 10 26 3'10 12'11 777
13 0 10 27 3'11 5'10 9'10
14 0 10 27 3'12 5'11 99
15 0 10 28 3'13 5'14 2'8'12
16 0 10 28 3'14 5'15 11'11
17 0 10 29 3'15 5'14 9'15
18 0 10 29 3'15 5'14 8'13
19 0 10 2'10 3'16 5'15 10'11
20 0 10 2'10 3'17 6'13 7'19
21 0 10 2'11 3'18 6'17 10'20
22 0 10 2'11 3'19 5'17 9'17
23 0 10 2'12 3'20 5'18 7'19
24 0 10 2'12 3'20 4'23 7'17

There is a huge online-archive of integer sequenzes: The On-Line Encyclopedia of Integer Sequences. The following A-numbers link to entries in that database.
For example A064867 the sequenz of the smallest numbers with persitence 3 or A064868 the sequenz of the smallest numbers with persistence 4.

Sequence of min(b,5) A064869 download: (txt)
Sequenz of min(b,6) (large) A064870 download: (txt)
Sequenz of min(b,7) (large) (big) A064871 download: (txt)
Sequenz of min(b,8) (large) (big) A064872 download: (txt)
Reihe von min(b,9) download: (txt)

3. Bounds on pmax
Last Update: by Sascha Kurz
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