A266946 Smallest number of each digital type.

1, 10, 11, 100, 101, 102, 110, 111, 1000, 1001, 1002, 1010, 1011, 1012, 1020, 1021, 1022, 1023, 1100, 1101, 1102, 1110, 1111, 10000, 10001, 10002, 10010, 10011, 10012, 10020, 10021, 10022, 10023, 10100, 10101, 10102, 10110, 10111, 10112, 10120, 10121, 10122, 10123, 10200, 10201, 10202, 10203, 10210, 10211, 10212, 10213, 10220, 10221, 10222, 10223, 10230, 10231, 10232, 10233, 10234, 11000, 11001, 11002, 11010, 11011, 11012, 11020, 11021, 11022, 11023, 11100, 11101, 11102, 11110, 11111

Offset: 1

Author

Vladimir Shevelev, Jan 06 2016

Keywords

• nonn
• base

Comments

The smallest single-digit positive number is 1. This is the first type.

The smallest of the two-digit positive numbers with distinct digits is 10. This is the second type. The smallest of two-digit positive numbers with equal digits is 11. This is the third type, etc.

A digital type is an equivalence class of integers that share the same pattern of identical digits. a(n) defines a possible canonical form for this equivalence relation. It can be obtained from the distinct terms in A358497 after the following digit replacement: {1->1, 2->0, 3->2, 4->3, 5->4, 6->5, 7->6, 8->7, 9->8, 0->9}. - Dmytro Inosov, Nov 14 2024

Examples

			The first 3-digit number is 100 = a(4).
The following number is 101. It does not belong to the type 100, since the first and the third digits coincide in 101, while in 100 they do not. So 101 is a new type, and a(5)=101.
Next consider 102. Here there are 3 distinct digits, so 102 is a new type, and a(6)=102. However, 103, 104, 105, 106, 107, 108, 109 also have 3 distinct digits, i.e., they belong to type 102.
Further, 110 belongs to neither type 100 nor type 101, since in 110 the first and the second digits coincide, while not in 100 and 101, so a(7)=110; also 111 is a new type, where all digits coincide.
Now we see that every 3-digit number is of one of the 5 types a(4), a(5), a(6), a(7), a(8).
Next we consider the first 4-digit number a(9)=1000, etc.
		

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..5295
• Dmytro S. Inosov and Emil Vlasák, Cryptarithmically unique terms in integer sequences, arXiv:2410.21427 [math.NT], 2024. See pp. 3, 18.

Crossrefs

Cf. A164864, A264406, A358497

Formula

The number of distinct types of k-digit numbers equals A164864(k).

Extensions

More terms from Peter J. C. Moses, Jan 06 2016