OEIS/Harrows - Revision history

imported>Gfis at 06:26, 18 October 2020

2020-10-18T06:26:47Z

← Older revision		Revision as of 06:26, 18 October 2020
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	We will call such a modified covering system a '''harrow'''.		We will call such a modified covering system a '''harrow'''.
	An arithmetic progression with a residue class 0 mod k will start with k.		An arithmetic progression with a residue class 0 mod k will start with k.
	== Level 4 - positive integers ==		=== Level 4 ([https://oeis.org/A000027 A000027] - positive integers) ===
	* 1, 2, 3, 4, ... ~~(OEIS sequence A000027)~~		* 1, 2, 3, 4, ...
	* inverse permutation of itself		* inverse permutation of itself
	0 mod 1		0 mod 1
	== Level 3 ==		=== Level 3 ([https://oeis.org/A160016 A160016]) ===
	* (0,) 2, 1, 4, 6, 8, 3, 10, 12, 14, 5, 16, 18, 20, 7, ... ~~(A160016)~~		* (0,) 2, 1, 4, 6, 8, 3, 10, 12, 14, 5, 16, 18, 20, 7, ...
	* signature (0,0,0,2,0,0,0,-1)		* linear recurrence with signature (0,0,0,2,0,0,0,-1)
			* Blocks of 4:
	2 mod 6		2 mod 6
	1 mod 2		1 mod 2 (odd numbers)
	4 mod 6		4 mod 6
	0 mod 6		0 mod 6
	Inverse permutation:		==== Inverse permutation ([https://oeis.org/A338206 A338206]) ====
	* (0,) 2, 1, 6, 3, 10, 4, 14, 5, 18, 7, 22, 8, 26, ... ~~(A338206)~~		* (0,) 2, 1, 6, 3, 10, 4, 14, 5, 18, 7, 22, 8, 26, ...
	* signature (0,1,0,0,0,1,0,-1)		* linear recurrence with signature (0,1,0,0,0,1,0,-1)
			* Blocks of 6:
	2 mod 12		2 mod 12
	1 mod 4		1 mod 4
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	3 mod 4		3 mod 4
	10 mod 12		10 mod 12
	0 mod 4		4 mod 4

			=== Level 2 ([https://oeis.org/A307048 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 ([https://oeis.org/A338207 A338207]) ====
			* 2, 1, 14, 6, 4, 3, 8, 12, 20, 5, 122, 18, 10, 7, 50, 24, 38, ...

			=== Level 1 ([https://oeis.org/A322469 A322469]) ===
			* 3, 1, 2, 7, 11, 15, 5, 10, 19, 23, 27, 9, 18, 6, 12, 4, 8, ...
			==== Inverse permutation ([https://oeis.org/A338208 A338208]) ====
			* 2, 3, 1, 16, 7, 14, 4, 17, 12, 8, 5, 15, 21, 33, 6, 126, 26, 13, 9, ...

imported>Gfis: Level 4, 3

2020-10-17T20:53:09Z

Level 4, 3

@@ Line 5: / Line 5: @@
 In the following we will use a slightly different definition:
-# negative integers are excluded,
+# negative integers and zero are excluded,
 # there may be infinitely many residue classes, and
-# any non-negative integer is covered by exactly one residue class only.
+# any positive integer is covered by exactly one residue class only,
 We will call such a modified covering system a '''harrow'''.

imported>Gfis: Created page with "In the early 1930s [https://en.wikipedia.org/wiki/Paul_Erd%C5%91s Erdős] introduced a [https://en.wikipedia.org/wiki/Covering_system covering system] (also called '''complete..."

2020-10-17T20:04:09Z

Created page with "In the early 1930s [https://en.wikipedia.org/wiki/Paul_Erd%C5%91s Erdős] introduced a [https://en.wikipedia.org/wiki/Covering_system covering system] (also called '''complete..."

New page

In the early 1930s [https://en.wikipedia.org/wiki/Paul_Erd%C5%91s Erdős] introduced a
[https://en.wikipedia.org/wiki/Covering_system covering system] (also called '''complete residue system'''s) as a collection
:<math>\{a_1(\mathrm{mod}\ {n_1}),\ \ldots,\ a_k(\mathrm{mod}\ {n_k})\}</math>
of finitely many [[residue class]]es <math> a_i(\mathrm{mod}\ {n_i}) = \{ a_i + n_ix:\ x \in \Z \} </math>

In the following we will use a slightly different definition:
# negative integers are excluded,
# there may be infinitely many residue classes, and
# any non-negative integer is covered by exactly one residue class only.
We will call such a modified covering system a '''harrow'''.