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.

 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
 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.