cp's OEIS Frontend

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

%I A329300 #4 Nov 18 2019 22:09:21
%S A329300 1,26,343,9374,3203,411942,1203135,12555566,23577999,475857425,
%T A329300 78497711,1840723325,44509735045,11166989789,9181683711,
%U A329300 1240214273284785,93417582527,538955006315,81324126339,123196100516,3851792910943,5652942368056,4967531840023463
%N A329300 a(n) is (apparently) the largest number whose digits are in nondecreasing order in bases n and n+1.
%C A329300 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.
%e A329300 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.
%e A329300 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.
%e A329300 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.
%e A329300 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):
%e A329300 .
%e A329300    n |  a(n) in base 10 |  a(n) in base n  | a(n) in base n+1
%e A329300   ---+------------------+------------------+-----------------
%e A329300    2 |                1 |             1_2  |             1_3
%e A329300    3 |               26 |           222_3  |           122_4
%e A329300    4 |              343 |         11113_4  |          2333_5
%e A329300    5 |             9374 |        244444_5  |        111222_6
%e A329300    6 |             3203 |         22455_6  |         12224_7
%e A329300    7 |           411942 |       3333666_7  |       1444446_8
%e A329300    8 |          1203135 |       4455677_8  |       2233346_9
%e A329300    9 |         12555566 |      25555888_9  |      12555566_10
%e A329300   10 |         23577999 |      23577999_10 |      12344555_11
%e A329300   11 |        475857425 |     2246777aa_11 |     113444555_12
%e A329300   12 |         78497711 |      22356abb_12 |      13355666_13
%e A329300   13 |       1840723325 |     23447abcc_13 |     136677777_14
%e A329300   14 |      44509735045 |    22234ccccd_14 |    125789999a_15
%e A329300   15 |      11166989789 |     455577aae_15 |     2999abddd_16
%e A329300   16 |       9181683711 |     223455fff_16 |     1566aadee_17
%e A329300   17 | 1240214273284785 | 223333444588g_17 | 115669aaaffff_18
%e A329300   18 |      93417582527 |     88aaabhhh_18 |     599cdeefg_19
%e A329300   19 |     538955006315 |    1cdhhhiiii_19 |    111138ffff_20
%e A329300   20 |      81324126339 |     33adfffgj_20 |     23347ffff_21
%e A329300   21 |     123196100516 |     3588ghjkk_21 |     258cfffgg_22
%e A329300   22 |    3851792910943 |    34449ijlll_22 |    234677888c_23
%e A329300   23 |    5652942368056 |    33466ikmmm_23 |    238ceefffg_24
%e A329300   24 | 4967531840023463 |  3688bdfkkmmn_24 |  22255aaabcdd_25
%Y A329300 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).
%K A329300 nonn,base
%O A329300 2,2
%A A329300 _Jon E. Schoenfield_, Nov 17 2019