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%I #90 Jul 29 2019 13:33:28
%S 1,3,1,5,11,1,7,29,59,1,9,55,29,59,1,11,89,119,1109,827,1,13,131,289,
%T 3703,3701,2875,1,15,181,563,5289,7799,34805,15163,1,17,239,965,16115,
%U 45289,138871,128117,31547,1,19,305,1519,25661,57587,745289,1711735,687989,97083,1
%N Square array A(m,n) (m>=1, n>=1) read by antidiagonals: A(m,n) = (2*n - 1)^^m mod (2*n)^m (see Comments for definition of ^^).
%C Tetration (x^^n) is defined as x^^0 = 1 and x^^n = x^(x^^(n - 1)). Another way to put this is that x^^n = x^x^x^...x with n x's.
%C Conjecture: For any three integers (greater than 1), m, n, and k, such that (2*n - 1)^^m == k (mod (2*n)^m), (2*n - 1)^k == k (mod (2*n)^m). For example, 5^^2 == 29 (mod 6^2) and 5^29 == 29 (mod 6^2).
%C Conjecture: For n > 1 and m >= 2, floor(((2*n - 1)^^m)/(2*n)) == 2*(n - 1) (mod 2*n). For example, floor((13^^3)/14) == 12 (mod 14) and floor((15^^4)/16) == 14 (mod 16).
%C Conjecture: For m > 1, where (2*n - 1)^^m == j (mod (2*n)^(m + 1)), A(m + 1,n) = j. For example, A(6,3) = 563 and A(6,4) = 16115; 11^^3 == 563 (mod 12^3) and 11^^3 == 16115 (mod 12^4).
%H Charles W. Trigg, <a href="https://cms.math.ca/crux/backfile/Crux_v7n06_Jun.pdf">Problem 559</a>, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.
%H Eric Weisstein's World of Mathematics,<a href="http://mathworld.wolfram.com/PowerTower.html">Power Tower</a>.
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Tetration">Tetration</a>.
%e Square array A(m,n) begins:
%e \n 1 2 3 4 5 6 7 8 ...
%e m\
%e 1| 1 3 5 7 9 11 13 15 ...
%e 2| 1 11 29 55 89 131 181 239 ...
%e 3| 1 59 29 119 289 563 965 1519 ...
%e 4| 1 59 1109 3703 5289 16115 25661 13807 ...
%e 5| 1 827 3701 7799 45289 57587 332989 669167 ...
%e 6| 1 2875 34805 138871 745289 1799411 4635581 669167 ...
%e 7| 1 15163 128117 1711735 2745289 25687283 49812797 67778031 ...
%e 8| 1 31547 687989 8003191 92745289 419837171 155226301 3557438959 ...
%e .
%e Examples of columns in this array:
%e A(m,1) = A000012(m - 1).
%e A(m,5) = A306686(m) with a note about how this sequence repeats terms rather than skipping.
%e Examples of rows in this array:
%e A(1,n) = A005408(n - 1).
%e A(2,n) = A082108(n - 1).
%o (PARI) tetrmod(b,n,m)=my(t=b);i=0;while(i<=n, i++&&if(i>1, t=lift(Mod(b,m)^t), t)); t
%o tetrmatrix(lim)= matrix(lim,lim,x,y,tetrmod((2*y)-1,x,(2*y)^x))
%Y Cf. A000012, A005408, A082108, A306686.
%K nonn,tabl
%O 1,2
%A _Davis Smith_, Mar 28 2019
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