A329298 -id:A329298 - cp's OEIS frontend

A329294 Numbers whose digits are in nondecreasing order in bases 4 and 5.

0, 1, 2, 3, 6, 7, 31, 43, 63, 343

Offset: 1

Jon E. Schoenfield, Nov 09 2019

There are no more terms through 10^10000 (which is a 16610-digit number in base 4 and a 14307-digit number in base 5). But can it be proved that 343 is the final term of the sequence?

			a(1)  =   0 =     0_4 =    0_5
a(2)  =   1 =     1_4 =    1_5
a(3)  =   2 =     2_4 =    2_5
a(4)  =   3 =     3_4 =    3_5
a(5)  =   6 =    12_4 =   11_5
a(6)  =   7 =    13_4 =   12_5
a(7)  =  31 =   133_4 =  111_5
a(8)  =  43 =   223_4 =  133_5
a(9)  =  63 =   333_4 =  223_5
a(10) = 343 = 11113_4 = 2333_5

Intersection of A023746 (base 4) and A023747 (base 5).

Numbers whose digits are in nondecreasing order in bases b and b+1: this sequence (b=4), A329295 (b=5), A329296 (b=6), A329297 (b=7), A329298 (b=8), A329299 (b=9). See A329300 for the (apparently) largest term of each of these sequences.

isnondec(v) = (#v==0) || (#select(x->(x<0), vector(#v-1, k, v[k+1]-v[k])) == 0);
isok(n) = isnondec(digits(n, 4)) && isnondec(digits(n, 5)); \\ Michel Marcus, Nov 11 2019

A329295 Numbers whose digits are in nondecreasing order in bases 5 and 6.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 8, 9, 14, 43, 44, 64, 93, 94, 784, 1562, 1563, 1564, 1569, 1599, 3124, 9374

Offset: 1

Jon E. Schoenfield, Nov 17 2019

There are no more terms through 10^10000 (which is a 14307-digit number in base 5 and a 12851-digit number in base 6). But can it be proved that 9374 is the final term of the sequence?

			a(1)  =    0 =      0_5 =      0_6
a(2)  =    1 =      1_5 =      1_6
a(3)  =    2 =      2_5 =      2_6
a(4)  =    3 =      3_5 =      3_6
a(5)  =    4 =      4_5 =      4_6
a(6)  =    7 =     12_5 =     11_6
a(7)  =    8 =     13_5 =     12_6
a(8)  =    9 =     14_5 =     13_6
a(9)  =   14 =     24_5 =     22_6
a(10) =   43 =    133_5 =    111_6
a(11) =   44 =    134_5 =    112_6
a(12) =   64 =    224_5 =    144_6
a(13) =   93 =    333_5 =    233_6
a(14) =   94 =    334_5 =    234_6
a(15) =  784 =  11114_5 =   3344_6
a(16) = 1562 =  22222_5 =  11122_6
a(17) = 1563 =  22223_5 =  11123_6
a(18) = 1564 =  22224_5 =  11124_6
a(19) = 1569 =  22234_5 =  11133_6
a(20) = 1599 =  22344_5 =  11223_6
a(21) = 3124 =  44444_5 =  22244_6
a(22) = 9374 = 244444_5 = 111222_6

Intersection of A023747 (base 5) and A023748 (base 6).

Numbers whose digits are in nondecreasing order in bases b and b+1: A329294 (b=4), this sequence (b=5), A329296 (b=6), A329297 (b=7), A329298 (b=8), A329299 (b=9). See A329300 for the (apparently) largest term of each of these sequences.

A329296 Numbers whose digits are in nondecreasing order in bases 6 and 7.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 8, 9, 10, 11, 16, 17, 57, 58, 59, 65, 89, 130, 131, 172, 173, 179, 1600, 1601, 3203

Offset: 1

Jon E. Schoenfield, Nov 17 2019

There are no more terms through 10^10000 (which is a 12851-digit number in base 6 and an 11833-digit number in base 7). But can it be proved that 3203 is the final term of the sequence?

			a(1)  =    0 =     0_6 =     0_7
a(2)  =    1 =     1_6 =     1_7
a(3)  =    2 =     2_6 =     2_7
a(4)  =    3 =     3_6 =     3_7
a(5)  =    4 =     4_6 =     4_7
a(6)  =    5 =     5_6 =     5_7
a(7)  =    8 =    12_6 =    11_7
a(8)  =    9 =    13_6 =    12_7
a(9)  =   10 =    14_6 =    13_7
a(10) =   11 =    15_6 =    14_7
a(11) =   16 =    24_6 =    22_7
a(12) =   17 =    25_6 =    23_7
a(13) =   57 =   133_6 =   111_7
a(14) =   58 =   134_6 =   112_7
a(15) =   59 =   135_6 =   113_7
a(16) =   65 =   145_6 =   122_7
a(17) =   89 =   225_6 =   155_7
a(18) =  130 =   334_6 =   244_7
a(19) =  131 =   335_6 =   245_7
a(20) =  172 =   444_6 =   334_7
a(21) =  173 =   445_6 =   335_7
a(22) =  179 =   455_6 =   344_7
a(23) = 1600 = 11224_6 =  4444_7
a(24) = 1601 = 11225_6 =  4445_7
a(25) = 3203 = 22455_6 = 12224_7

Intersection of A023748 (base 6) and A023749 (base 7). Numbers whose digits are in nondecreasing order in bases b and b+1: A329294 (b=4), A329295 (b=5), this sequence (b=6), A329297 (b=7), A329298 (b=8), A329299 (b=9). See A329300 for the (apparently) largest term of each of these sequences.

A329297 Numbers whose digits are in nondecreasing order in bases 7 and 8.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 9, 10, 11, 12, 13, 18, 19, 20, 27, 73, 74, 75, 76, 82, 83, 118, 146, 173, 174, 228, 229, 230, 237, 293, 587, 685, 804, 2925, 14062, 42131, 42132, 42139, 411942

Offset: 1

Jon E. Schoenfield, Nov 17 2019

There are no more terms through 10^10000 (which is an 11833-digit number in base 7 and an 11074-digit number in base 8). But can it be proved that 411942 is the final term of the sequence?

			Sequence includes 7 terms that are 1-digit numbers in both bases, 9 terms that are 2-digit numbers in both bases, and the following:
  a(17) =     73 =     133_7 =     111_8
  a(18) =     74 =     134_7 =     112_8
  a(19) =     75 =     135_7 =     113_8
  a(20) =     76 =     136_7 =     114_8
  a(21) =     82 =     145_7 =     122_8
  a(22) =     83 =     146_7 =     123_8
  a(23) =    118 =     226_7 =     166_8
  a(24) =    146 =     266_7 =     222_8
  a(25) =    173 =     335_7 =     255_8
  a(26) =    174 =     336_7 =     256_8
  a(27) =    228 =     444_7 =     344_8
  a(28) =    229 =     445_7 =     345_8
  a(29) =    230 =     446_7 =     346_8
  a(30) =    237 =     456_7 =     355_8
  a(31) =    293 =     566_7 =     445_8
  a(32) =    587 =    1466_7 =    1113_8
  a(33) =    685 =    1666_7 =    1255_8
  a(34) =    804 =    2226_7 =    1444_8
  a(35) =   2925 =   11346_7 =    5555_8
  a(36) =  14062 =   55666_7 =   33356_8
  a(37) =  42131 =  233555_7 =  122223_8
  a(38) =  42132 =  233556_7 =  122224_8
  a(39) =  42139 =  233566_7 =  122233_8
  a(40) = 411942 = 3333666_7 = 1444446_8

Intersection of A023749 (base 7) and A023750 (base 8). Numbers whose digits are in nondecreasing order in bases b and b+1: A329294 (b=4), A329295 (b=5), A329296 (b=6), this sequence (b=7), A329298 (b=8), A329299 (b=9). See A329300 for the (apparently) largest term of each of these sequences.

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

Original entry on oeis.org

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

Jon E. Schoenfield, Nov 17 2019

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.

			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

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).

cp's OEIS Frontend

A329294 Numbers whose digits are in nondecreasing order in bases 4 and 5.

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A329295 Numbers whose digits are in nondecreasing order in bases 5 and 6.

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A329296 Numbers whose digits are in nondecreasing order in bases 6 and 7.

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A329297 Numbers whose digits are in nondecreasing order in bases 7 and 8.

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

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