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%I #18 Dec 27 2017 09:49:35
%S 1,2,1,1,1,1,1,1,1,4,2,1,1,4,2,2,1,1,2,1,1,3,3,2,1,1,6,6,4,2,1,2,1,4,
%T 1,1,1,1,1,1,1,6,1,3,1,2,1,1,1,6,1,3,4,2,1,1,4,1,4,2,2,1,4,6,2,1,3,6,
%U 2,1,3,10,5,10,10,2,1,1,5,5,10,5,2,2,1,1,2,1,1,1,2,2,1,1,2,2,1,1,1,1,1
%N Irregular triangle read by rows. T(n, k) gives the period length of the periodic sequence {A290600(n, k)^i}_{i >= A290601(n, k)} (mod A002808(n)), for n >= 1 and k = 1..A290599(n).
%C The length of row n is A290599(n).
%C See A290601 for the proof that this sequence is defined, and the definition of the type of periodicity (imin,P) with imin = A290601(n, k) and the period length P = T(n, k).
%e The irregular triangle T(n, k) begins (N(n) = A002808(n)):
%e n N(n) \ k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e 1 4 1
%e 2 6 2 1 1
%e 3 8 1 1 1
%e 4 9 1 1
%e 5 10 4 2 1 1 4
%e 6 12 2 2 1 1 2 1 1
%e 7 14 3 3 2 1 1 6 6
%e 8 15 4 2 1 2 1 4
%e 9 16 1 1 1 1 1 1 1
%e 10 18 6 1 3 1 2 1 1 1 6 1 3
%e 11 20 4 2 1 1 4 1 4 2 2 1 4
%e 12 21 6 2 1 3 6 2 1 3
%e 13 22 10 5 10 10 2 1 1 5 5 10 5
%e 14 24 2 2 1 1 2 1 1 1 2 2 1 1 2 2 1
%e 15 25 1 1 1 1
%e ...
%e T(5, 1) = 4 because A290600(5, 1) = 2, N(5) = A002808(5) = 10, A290601(5, 1) = 1 and {2^i}_{i>=1} (mod 10) == {repeat(2,4,8,6)} with period length 4. This is of the type (1,4).
%e T(7, 6) = 6 because A290600(7, 6) = 10, N(7) = A002808(7) = 14, A290601(7, 6) = 1 and {10^i}_{i>=1} (mod 14) == {repeat(10, 2, 6, 4, 12, 8)} with period length 4. Type (1,6).
%e The sequence {A290600(10, 1)^i}_{i >= A290601(10, 1)} (mod A002808(10)) = {2^i}_{i >= 1} (mod 18) is periodic with period length P = T(10, 1) = 6. Namely, {repeat(2, 4, 8, 16, 14, 10)}, of type (1,6).
%e The periodicity types (imin,P) = (A290601(n, k), A290602(n, k)) begin:
%e n N(n) \ k 1 2 3 4 5 6 7 8 9 10 11
%e 1 4 (2,1)
%e 2 6 (1,2) (1,1) (1,1)
%e 3 8 (3,1) (2,1) (3,1)
%e 4 9 (2,1) (2,1)
%e 5 10 (1,4) (1,2) (1,1) (1,1) (1,4)
%e 6 12 (2,2) (1,2) (1,1) (2,1) (1,2) (1,1) (2,1)
%e 7 14 (1,3) (1,3) (1,2) (1,1) (1,1) (1,6) (1,6)
%e 8 15 (1,4) (1,2) (1,1) (1,2) (1,1) (1,4)
%e 9 16 (4,1) (2,1) (4,1) (2,1) (4,1) (2,1) (4,1)
%e 10 18 (1,6) (2,1) (1,3) (2,1) (1,2) (1,1) (1,1) (2,1) (1,6) (2,1) (1,3)
%e 11 20 (2,4) (1,2) (1,1) (2,1) (1,4) (2,1) (1,4) (2,2) (1,2) (1,1) (2,4)
%e 12 21 (1,6) (1,2) (1,1) (1,3) (1,6) (1,2) (1,1) (1,3)
%e 13 22 (1,10) (1,5) (1,10) (1,10) (1,2) (1,1) (1,1) (1,5) (1,5) (1,10) (1,5)
%e ...
%e ----------------------------------------------------------------------------------
%Y Cf. A002808, A290599, A290600, A290601.
%K nonn,tabf
%O 1,2
%A _Wolfdieter Lang_, Aug 30 2017
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