A329300 a(n) is (apparently) the largest number whose digits are in nondecreasing order in bases n and n+1.

1, 26, 343, 9374, 3203, 411942, 1203135, 12555566, 23577999, 475857425, 78497711, 1840723325, 44509735045, 11166989789, 9181683711, 1240214273284785, 93417582527, 538955006315, 81324126339, 123196100516, 3851792910943, 5652942368056, 4967531840023463

Offset: 2

Author

Jon E. Schoenfield, Nov 17 2019

Keywords

• nonn
• base

Comments

In the Data section, each value for n < 9 has been confirmed as the largest such number < 10^10000, and each of the remaining values has been confirmed as the largest such number < 10^1000.

Examples

			The only numbers whose digits are in nondecreasing order in base 2 are the numbers of the form 2^k-1 (k >= 0); of those, the only numbers whose digits are in nondecreasing order in base 3 are 0 = 0_2 = 0_3 and 1 = 1_2 = 1_3. The larger of these numbers is 1, so a(2) = 1.
Up to at least 10^10000, the only numbers whose digits are in nondecreasing order in both base 3 and base 4 are 0 = 0_3 = 0_4, 1 = 1_3 = 1_4, 2 = 2_3 = 2_4, 5 = 12_3 = 11_4, and 26 = 222_3 = 122_4. The largest of these numbers is 26, so a(3) = 26.
A329294 lists the numbers (up to at least 10^10000) whose digits are in nondecreasing order in both base 4 and base 5, the largest of which is 343, so a(4) = 343.
The following table lists the values of a(n) for n = 2..24 with their base-n and base-(n+1) expansions (where the letters a, b, c, etc. represent the digit values 10, 11, 12, etc., respectively):
.
   n |  a(n) in base 10 |  a(n) in base n  | a(n) in base n+1
  ---+------------------+------------------+-----------------
   2 |                1 |             1_2  |             1_3
   3 |               26 |           222_3  |           122_4
   4 |              343 |         11113_4  |          2333_5
   5 |             9374 |        244444_5  |        111222_6
   6 |             3203 |         22455_6  |         12224_7
   7 |           411942 |       3333666_7  |       1444446_8
   8 |          1203135 |       4455677_8  |       2233346_9
   9 |         12555566 |      25555888_9  |      12555566_10
  10 |         23577999 |      23577999_10 |      12344555_11
  11 |        475857425 |     2246777aa_11 |     113444555_12
  12 |         78497711 |      22356abb_12 |      13355666_13
  13 |       1840723325 |     23447abcc_13 |     136677777_14
  14 |      44509735045 |    22234ccccd_14 |    125789999a_15
  15 |      11166989789 |     455577aae_15 |     2999abddd_16
  16 |       9181683711 |     223455fff_16 |     1566aadee_17
  17 | 1240214273284785 | 223333444588g_17 | 115669aaaffff_18
  18 |      93417582527 |     88aaabhhh_18 |     599cdeefg_19
  19 |     538955006315 |    1cdhhhiiii_19 |    111138ffff_20
  20 |      81324126339 |     33adfffgj_20 |     23347ffff_21
  21 |     123196100516 |     3588ghjkk_21 |     258cfffgg_22
  22 |    3851792910943 |    34449ijlll_22 |    234677888c_23
  23 |    5652942368056 |    33466ikmmm_23 |    238ceefffg_24
  24 | 4967531840023463 |  3688bdfkkmmn_24 |  22255aaabcdd_25
		

Crossrefs

Numbers whose digits are in nondecreasing order in bases b and b+1: A329294 (b=4), A329295 (b=5), A329296 (b=6), A329297 (b=7), A329298 (b=8), A329299 (b=9).