cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A329299 Numbers whose digits are in nondecreasing order in bases 9 and 10.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 11, 12, 13, 14, 15, 16, 17, 22, 23, 24, 25, 26, 33, 34, 35, 44, 111, 112, 113, 114, 115, 116, 122, 123, 124, 125, 133, 134, 188, 222, 223, 224, 233, 277, 278, 366, 367, 368, 377, 455, 456, 457, 458, 466, 467, 556, 557, 566, 1113
Offset: 1

Views

Author

Jon E. Schoenfield, Nov 17 2019

Keywords

Comments

a(91) = 12555566 is the largest term < 10^10000 (which is a 10480-digit number in base 9). But can it be proved that 12555566 is the final term of the sequence?

Examples

			Sequence includes, respectively, 9, 16, 32, and 11 terms that are 1-, 2-, 3-, and 4- digit terms in both bases, and the following:
  a(69) =    14777 =    22238_9
  a(70) =    15677 =    23448_9
  a(71) =    22234 =    33444_9
  a(72) =    22235 =    33445_9
  a(73) =    22236 =    33446_9
  a(74) =    22237 =    33447_9
  a(75) =    22238 =    33448_9
  a(76) =    22244 =    33455_9
  a(77) =    22245 =    33456_9
  a(78) =    22246 =    33457_9
  a(79) =    22247 =    33458_9
  a(80) =    22255 =    33467_9
  a(81) =    22256 =    33468_9
  a(82) =    22335 =    33566_9
  a(83) =    22336 =    33567_9
  a(84) =    22337 =    33568_9
  a(85) =    22345 =    33577_9
  a(86) =    22346 =    33578_9
  a(87) =    22355 =    33588_9
  a(88) =    44468 =    66888_9
  a(89) =   222344 =   367888_9
  a(90) =  1233467 =  2278888_9
  a(91) = 12555566 = 25555888_9

Crossrefs

Intersection of A023751 (base 9) and A009994 (base 10). 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), A329299 (b=8), this sequence (b=9). See A329300 for the (apparently) largest term of each of these sequences.

Programs

  • Maple
    filter:= proc(n) local L;
  L:= convert(n,base,10);
  `and`(seq(L[i+1]<=L[i],i=1..nops(L)-1))
end proc:
ND[1]:= [$1..8]: R:= $0..8:
for d from 2 to 10 do
  ND[d]:= map(t -> seq(9*t+r, r=(t mod 9) ..8), ND[d-1]);
  R:= R, op(select(filter, ND[d]));
od:
R; # Robert Israel, Nov 20 2019
  • Mathematica
    Select[Range[0,1200],Min[Differences[IntegerDigits[#]]]>-1&& Min[ Differences[ IntegerDigits[ #,9]]]>-1&] (* Harvey P. Dale, Oct 14 2022 *)

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

Original entry on oeis.org

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

Views

Author

Jon E. Schoenfield, Nov 09 2019

Keywords

Comments

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?

Examples

			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

Crossrefs

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.

Programs

  • PARI
    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

Views

Author

Jon E. Schoenfield, Nov 17 2019

Keywords

Comments

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?

Examples

			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

Crossrefs

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.

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

Views

Author

Jon E. Schoenfield, Nov 17 2019

Keywords

Comments

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?

Examples

			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

Crossrefs

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

Views

Author

Jon E. Schoenfield, Nov 17 2019

Keywords

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).

A329298 Numbers whose digits are in nondecreasing order in bases 8 and 9.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 10, 11, 12, 13, 14, 15, 20, 21, 22, 23, 30, 31, 91, 92, 93, 94, 95, 101, 102, 103, 111, 151, 182, 183, 222, 223, 293, 294, 295, 303, 365, 366, 367, 374, 375, 822, 823, 831, 951, 1023, 10023, 14774, 14775, 14783, 599551, 608623, 1203126, 1203127, 1203135
Offset: 1

Views

Author

Jon E. Schoenfield, Nov 17 2019

Keywords

Comments

There are no more terms through 10^10000 (which is an 11074-digit number in base 8 and a 10480-digit number in base 9). But can it be proved that 1203135 is the final term of the sequence?

Examples

			Sequence includes 8 terms that are 1-digit numbers in both bases, 12 that are 2-digit numbers in both bases, 23 that are 3-digit terms in both bases, and the following:
  a(44) =     822 =    1466_8 =    1113_9
  a(45) =     823 =    1467_8 =    1114_9
  a(46) =     831 =    1477_8 =    1123_9
  a(47) =     951 =    1667_8 =    1266_9
  a(48) =    1023 =    1777_8 =    1356_9
  a(49) =   10023 =   23447_8 =   14666_9
  a(50) =   14774 =   34666_8 =   22235_9
  a(51) =   14775 =   34667_8 =   22236_9
  a(52) =   14783 =   34677_8 =   22245_9
  a(53) =  599551 = 2222777_8 = 1113377_9
  a(54) =  608623 = 2244557_8 = 1126777_9
  a(55) = 1203126 = 4455666_8 = 2233336_9
  a(56) = 1203127 = 4455667_8 = 2233337_9
  a(57) = 1203135 = 4455677_8 = 2233346_9

Crossrefs

Intersection of A023750 (base 8) and A023751 (base 9). 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), this sequence (b=8), A329299 (b=9). See A329300 for the (apparently) largest term of each of these sequences.
Showing 1-6 of 6 results.

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