OEIS/Harrows

In the early 1930s Erdős introduced a covering system (also called complete residue systems) as a collection

$\{a_{1}(\mathrm {mod} \ {n_{1}}),\ \ldots ,\ a_{k}(\mathrm {mod} \ {n_{k}})\}$

of finitely many residue classes $a_{i}(\mathrm {mod} \ {n_{i}})=\{a_{i}+n_{i}x:\ x\in \mathbb {Z} \}$

In the following we will use a slightly different definition:

1. negative integers and zero are excluded,
2. there may be infinitely many residue classes, and
3. any positive integer is covered by exactly one residue class only,
4. the residue classes that define the arithmetic progressions are ordered, and this order defines the permutation of the integers.

We will call such a modified covering system a harrow. An arithmetic progression with a residue class 0 mod k will start with k.

Level 4 (A000027 - positive integers)

• 1, 2, 3, 4, ...
• inverse permutation of itself

 0 mod 1

Level 3 (A160016)

• (0,) 2, 1, 4, 6, 8, 3, 10, 12, 14, 5, 16, 18, 20, 7, ...
• linear recurrence with signature (0,0,0,2,0,0,0,-1)
• Blocks of 4:

 2 mod 6
 1 mod 2 (odd numbers)
 4 mod 6
 0 mod 6

Inverse permutation (A338206)

• (0,) 2, 1, 6, 3, 10, 4, 14, 5, 18, 7, 22, 8, 26, ...
• linear recurrence with signature (0,1,0,0,0,1,0,-1)
• Blocks of 6:

 2 mod 12
 1 mod  4
 6 mod 12
 3 mod  4
10 mod 12
 4 mod  4

Level 2 (A307048)

• 2, 1, 6, 5, 10, 4, 14, 7, 18, 13, 22, 8, 26, 3, 30, 21, 34, 12, 38, ...

 2 mod 4
 4 mod 4
 5 mod 8
 1 mod 8
 7 mod 16
15 mod 16
19 mod 32

Inverse permutation (A338207)

• 2, 1, 14, 6, 4, 3, 8, 12, 20, 5, 122, 18, 10, 7, 50, 24, 38, ...

Level 1 (A322469)

• 3, 1, 2, 7, 11, 15, 5, 10, 19, 23, 27, 9, 18, 6, 12, 4, 8, ...

Inverse permutation (A338208)

• 2, 3, 1, 16, 7, 14, 4, 17, 12, 8, 5, 15, 21, 33, 6, 126, 26, 13, 9, ...