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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):
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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
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