A203575 Array of certain four complete residue classes (nonnegative members), read by SW-NE antidiagonals.

0, 1, 4, 2, 7, 8, 3, 6, 9, 12, 5, 10, 15, 16, 11, 14, 17, 20, 13, 18, 23, 24, 19, 22, 25, 28, 21, 26, 31, 32, 27, 30, 33, 36, 29, 34, 39, 40, 35, 38, 41, 44, 37, 42, 47, 48, 43, 46

Offset: 1

Author

Wolfdieter Lang, Jan 12 2012

Keywords

• nonn
• tabf
• easy

Comments

See A193682 for the sequence called P_4, with period length 8, which defines the four complete residue classes [m], m = 0,1,2,3, via the equivalence relation p==q iff P_4(p) = P_4(q).

See a comment on A203571 for the general P_k sequences, and the multiplicative (but not additive) structure of these residue classes.

The row length sequence of this tabf array is [1,2,3,4,4,4,...].

This array defines a certain permutation of the nonnegative integers.

Examples

			The array starts
n\m  1   2   3   4
1:   0
2:   1   4
3:   2   7   8
4:   3   6   9  12
5:   5  10  15  16
6:  11  14  17  20
7:  13  18  23  24
8:  19  22  25  28
9:  21  26  31  32
10: 27  30  33  36
...
The sequence P_4(n)=A193682(n), n>=0, is repeated 0, 1, 2, 3, 0, 3, 2, 1, with period length 8. P_4(6)=2, hence 6 belongs to class [2].
Multiplicative structure: 11*23 == 3*1 = 3. Indeed: P_4(11*23) = P_4(253) = P_(5), because 253==5(mod 8), and P_(5)= 3, hence 11*23 belongs to class 3. In general, P_4(p*q) = P_4(P_4(p)*P_4(q)).
		

Crossrefs

Cf.A193682, A088520 (k=3), A090298 (k=5), A092260 (k=6), A113807 (k=7).

Formula

The nonnegative members of the four complete residue classes are (see a comment above for their definition):

[0]: 0, 4, 8, 12, 16, 20, 24, 28, 32, 36,... (A008586)

[1]: 1, 7, 9, 15, 17, 23, 25, 31, 33, 39,... (A047522)

[2]: 2, 6, 10, 14, 18, 22, 26, 30, 34, 38,... (A016825)

[3]: 3, 5, 11, 13, 19, 21, 27, 29, 35, 37,... (A047621)

In each class the corresponding negative numbers should be included.