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 1-3 of 3 results.

A266946 Smallest number of each digital type.

Original entry on oeis.org

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

Views

Author

Vladimir Shevelev, Jan 06 2016

Keywords

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

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

A267013 Number of distinct digital types of n-digit primes in base 10.

Original entry on oeis.org

1, 2, 4, 11, 51, 177, 876, 3965, 20782, 114459, 678536, 4160910, 27640731
Offset: 1

Views

Author

Vladimir Shevelev and Peter J. C. Moses, Jan 08 2016

Keywords

Comments

The sequence is related to A266991.
Sequence {A164864(n) - a(n)}_(n>=1) begins 0,0,1,4,1,26,1,175,365,1516,...
One can explain, why, for example, a(4)=11, instead of A164864(4)=15. There exist exactly 4 types of 4-digit numbers, which cannot be prime. In A266946 these types are: 1001, 1010, 1100, 1111. Indeed, numbers abba,aabb,aaaa are divisible by 11; a number abab is divisible by 101.
In other cases of n-digit types we should verify the divisibility of numbers of types in A266946 at least by primes of the form 11,101,... Besides, a digital type 1...1 exists only for n in A004023, i.e., for only 9 values of n from the first 270343. This simplifies the calculations.
a(n) <= A376918(n) with equality for n <= 9, but thereafter some digital types which pass the divisibility rules of A376918 don't in fact occur among the primes (see A377727). - Dmytro Inosov, Nov 05 2024
Based on the conjectured terms in A377727, the next three terms can be conjectured: a(14)=190402538; a(15)=1378294708; a(16)=10437142874. - Dmytro Inosov, Jan 07 2025

Links

Crossrefs

Cf. A164864, A164904, A264406, A266946, A376918, A377727.

Formula

a(n) = A376918(n) - A377727(n). - Dmytro Inosov, Nov 05 2024

Extensions

a(11)-a(13) from Michael S. Branicky, Nov 04 2024

A266991 Smallest representatives of primes of distinct digital types.

Original entry on oeis.org

2, 11, 13, 101, 103, 113, 199, 1009, 1013, 1021, 1033, 1039, 1103, 1117, 1151, 1303, 1511, 1777, 10007, 10037, 10061, 10099, 10103, 10111, 10133, 10139, 10141, 10211, 10223, 10243, 10271, 10301, 10303, 10313, 10331, 10333, 10343, 10399, 10513, 10607, 11003
Offset: 1

Views

Author

Vladimir Shevelev, Jan 08 2016

Keywords

Comments

Numbers of different digital types for n-digit primes are 1,2,4,11,...

Examples

			The first 3-digit prime is 101=a(4).
The following 3-digit prime is 103. It does not have the same digital type as 101, since in 103 there are 3 distinct digits, but not in 101. So a(5)=103.
The next two primes 107 and 109 belongs to type 103.
Next consider 113. Here the first two digits are the same, but in 101 and 103 they are not. So 113 is a new type, and a(6)=113.
It remains to find the smallest prime of form XYY. It is 199=a(7).
Now we see that every 3-digit prime is of one of the 4 types a(4),a(5),a(6),a(7).
Next we consider the first 4-digit number a(8)=1009, etc.

Links

Crossrefs

Cf. A264406, A266946.

Extensions

More terms from Peter J. C. Moses, Jan 08 2016
Showing 1-3 of 3 results.

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