Not Modest

> Hello SeqFans,

> Numbers which, when divided by their last digit, have their first digit as remainder:

>

> 13, 19, 23, 26, 29, 39, 46, 49, 59, 69, 79, 89, 103, 109, 127, ...

>

> Base-10 modest numbers (which are not exactly the same), can be found there: https://oeis.org/A054986

> Best,

> É.

Alois Heinz was quick to answer:

Hello Eric,

13, 19, 23, 26, 29, 39, 46, 49, 59, 69, 79, 89, 103, 109, 127, 133, 163, 193, 197, 199, 203, 206, 209, 214, 218, 233, 234, 236, 247, 254, 258, 263, 266, 274, 293, 294, 296, 298, 299, 309, 367, 399, 406, 409, 417, 428, 436, 466, 468, 487, 496, 499, 509, 537, 599, 609, 638, 657, 678, 699, 709, 799, 809, 899, 1003, 1009, 1033, 1037, 1063, 1093, 1099, 1107, 1123, 1153, 1177, 1183, 1189, 1213, 1243, 1247, 1273, 1279, 1303, 1317, 1333, 1363, 1369, 1387, 1393, 1423, 1453, 1457, 1459, 1483, 1513, 1527, 1543, 1549, 1573, 1597, ...

Best regards,

Alois

The next sequence asked would of course have this property:

> Hello SeqFans,

> Numbers which, when divided by their first digit, have their last digit as remainder:

>

> 10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61,

> 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85,

> 86, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, ...

>

> This is not a subsequence of https://oeis.org/A009995 ("Numbers with digits in strictly decreasing order").

>

> Best,

> É.

>

Again, Alois was quick:

10, 20, 21, 30, 31, 32, 40, 41, 42, 43, 50, 51, 52, 53, 54, 60, 61, 62, 63, 64, 65, 70, 71, 72, 73, 74, 75, 76, 80, 81, 82, 83, 84, 85, 86, 87, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 110, 120, 130, 140, 150, 160, 170, 180, 190, 200, 201, 210, 211, 220, 221, 230, 231, 240, 241, 250, 251, 260, 261, 270, 271, 280, 281, 290, 291, 300, 301, 302, 330, 331, 332, 360, 361, 362, 390, 391, 392, 400, 401, 402, 403, 420, 421, 422, 423, 440, 441, 442, 443, 460, ...

Best regards,

Alois

This is an automatic sequence in the terminology of Allouche & Shallit: it can be recognized by a regular expression (or finite-state machine) working on the decimal expansion of the number.

The easy parts:

1.*0

2.*[01]

4.*[02468][0123]

4[0123]

5.*[01234]

8 is routine but somewhat long.

3, 6, and 9 are longer than 8 but not too hard; here's 3:

3([0369]|[147][0369]*[258]|[147][0369]*[147][0369]*[147]|[258][0369]*[147])*[012]

7 is hard.  The length of the entire regular expression will be essentially the same as the length of this part, which will be several thousand characters.

Charles Greathouse

Analyst/Programmer

Case Western Reserve University

Maximilian Hasler has confirmed Alois’s first results and suggested the name “Restricted Modest Numbers” for those integers – good idea!

Many thanks to Alois, Charles and Maximilian!

Best,

É.