A324017 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 ^^).

1, 3, 1, 5, 11, 1, 7, 29, 59, 1, 9, 55, 29, 59, 1, 11, 89, 119, 1109, 827, 1, 13, 131, 289, 3703, 3701, 2875, 1, 15, 181, 563, 5289, 7799, 34805, 15163, 1, 17, 239, 965, 16115, 45289, 138871, 128117, 31547, 1, 19, 305, 1519, 25661, 57587, 745289, 1711735, 687989, 97083, 1

Offset: 1

Author

Davis Smith, Mar 28 2019

Keywords

• nonn
• tabl

Comments

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.

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).

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).

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).

Examples

			Square array A(m,n) begins:
  \n  1     2      3       4        5          6         7          8 ...
  m\
   1| 1     3      5       7        9         11        13         15 ...
   2| 1    11     29      55       89        131       181        239 ...
   3| 1    59     29     119      289        563       965       1519 ...
   4| 1    59   1109    3703     5289      16115     25661      13807 ...
   5| 1   827   3701    7799    45289      57587    332989     669167 ...
   6| 1  2875  34805  138871   745289    1799411   4635581     669167 ...
   7| 1 15163 128117 1711735  2745289   25687283  49812797   67778031 ...
   8| 1 31547 687989 8003191 92745289  419837171 155226301 3557438959 ...
.
Examples of columns in this array:
A(m,1) = A000012(m - 1).
A(m,5) = A306686(m) with a note about how this sequence repeats terms rather than skipping.
Examples of rows in this array:
A(1,n) = A005408(n - 1).
A(2,n) = A082108(n - 1).
		

Links

• Charles W. Trigg, Problem 559, Crux Mathematicorum, page 192, Vol. 7, Jun. 81.
• Eric Weisstein's World of Mathematics,Power Tower.
• Wikipedia, Tetration.

Crossrefs

Cf. A000012, A005408, A082108, A306686.

Programs

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
tetrmatrix(lim)= matrix(lim,lim,x,y,tetrmod((2*y)-1,x,(2*y)^x))