**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

**Charles Greathouse** adds this comment:

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.