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

%I A203575 #8 Mar 30 2012 18:49:34
%S A203575 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,
%T A203575 21,26,31,32,27,30,33,36,29,34,39,40,35,38,41,44,37,42,47,48,43,46
%N A203575 Array of certain four complete residue classes (nonnegative members), read by SW-NE antidiagonals.
%C A203575 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).
%C A203575 See a comment on A203571 for the general P_k sequences, and the multiplicative (but not additive) structure of these residue classes.
%C A203575 The row length sequence of this tabf array is [1,2,3,4,4,4,...].
%C A203575 This array defines a certain permutation of the nonnegative integers.
%F A203575 The nonnegative members of the four complete residue classes are (see a comment above for their definition):
%F A203575 [0]: 0, 4, 8, 12, 16, 20, 24, 28, 32, 36,...  (A008586)
%F A203575 [1]: 1, 7, 9, 15, 17, 23, 25, 31, 33, 39,...  (A047522)
%F A203575 [2]: 2, 6, 10, 14, 18, 22, 26, 30, 34, 38,... (A016825)
%F A203575 [3]: 3, 5, 11, 13, 19, 21, 27, 29, 35, 37,... (A047621)
%F A203575 In each class the corresponding negative numbers should be included.
%e A203575 The array starts
%e A203575 n\m  1   2   3   4
%e A203575 1:   0
%e A203575 2:   1   4
%e A203575 3:   2   7   8
%e A203575 4:   3   6   9  12
%e A203575 5:   5  10  15  16
%e A203575 6:  11  14  17  20
%e A203575 7:  13  18  23  24
%e A203575 8:  19  22  25  28
%e A203575 9:  21  26  31  32
%e A203575 10: 27  30  33  36
%e A203575 ...
%e A203575 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].
%e A203575 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)).
%Y A203575 Cf.A193682, A088520 (k=3), A090298 (k=5), A092260 (k=6), A113807 (k=7).
%K A203575 nonn,tabf,easy
%O A203575 1,3
%A A203575 _Wolfdieter Lang_, Jan 12 2012