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Array of certain four complete residue classes (nonnegative members), read by SW-NE antidiagonals.
0

%I #8 Mar 30 2012 18:49:34

%S 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 21,26,31,32,27,30,33,36,29,34,39,40,35,38,41,44,37,42,47,48,43,46

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

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

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

%C This array defines a certain permutation of the nonnegative integers.

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

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

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

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

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

%F In each class the corresponding negative numbers should be included.

%e The array starts

%e n\m 1 2 3 4

%e 1: 0

%e 2: 1 4

%e 3: 2 7 8

%e 4: 3 6 9 12

%e 5: 5 10 15 16

%e 6: 11 14 17 20

%e 7: 13 18 23 24

%e 8: 19 22 25 28

%e 9: 21 26 31 32

%e 10: 27 30 33 36

%e ...

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

%K nonn,tabf,easy

%O 1,3

%A _Wolfdieter Lang_, Jan 12 2012