OFFSET
1,3
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.
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.
EXAMPLE
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
KEYWORD
nonn,tabf,easy
AUTHOR
Wolfdieter Lang, Jan 12 2012
STATUS
approved