A085119 -id:A085119 - cp's OEIS frontend

A352196 a(n) = number of steps for the standard mod-n Ackermann function to stabilize to a set consisting of only one value, or -1 if it does not stabilize.

0, 2, 4, 3, 5, 4, 6, 3, 6, 5, 6, 4, 6, 4, 4, 4, 6, 4, 7, 4, 4, 5, 6, 4, 8, 4, 6, 4, 7, 4, 6, 5, 7, 6, 4, 4, 7, 6, 4, 4, 7, 4, 5, 5, 4, 5, 6, 4, 6, 5, 5, 4, 9, 5, 8, 4, 6, 6, 6, 4, 7, 5, 4, 5, 4, 5, 8, 5, 8, 4, 7, 4, 6, 6, 7, 6, 7, 4, 7, 4, 9, 6, 8, 4, 5, 5, 7, 5, 9, 4, 4, 5, 5, 5, 6, 5, 5, 6, 6, 5, 8, 5, 7, 4, 4, 6, 5, 5, 8, 6, 7, 4, 8, 5, 7, 5, 4, 7, 6

Offset: 1

N. J. A. Sloane, Mar 23 2022

nonn

This was Stan Wagon's Problem of the Week #1340, from March 2022, which in turn was based on a 1993 Monthly problem of Jon Froemke and Jerrold Grossman.
Stan Wagon mentions that Mark Rickert has found the first 8 million terms (see link), and the only one that does not stabilize is n = 1969 where it becomes periodic with period 2 after 8 steps. So a(1969) = -1.
[Needs program(s), b-file. - N. J. A. Sloane, May 25 2025]

Jon Froemke and Jerrold W. Grossman, A Mod-n Ackermann Function, or What's So Special About 1969?, The American Mathematical Monthly, Vol. 100, No. 2 (February 1993), pp. 180-183; ResearchGate link.
Mark Rickert, The first 8 million terms of A085119 [a gzipped file], March 2022.
Stan Wagon, Problem of the Week POW #1340: Modular Ackermann, March 2022.
Stan Wagon, Problem of the Week POW #1340: Solution, March 2022.

Cf. A085119.

"Standard" added to definition by N. J. A. Sloane, May 25 2025 to be consistent with the Froemke-Grossman (1993) article.

A383460 13 X oo array read by antidiagonals, giving the values of the standard mod 13 Ackermann function.

Original entry on oeis.org

1, 2, 2, 3, 3, 3, 5, 5, 4, 4, 0, 0, 7, 5, 5, 5, 5, 3, 9, 6, 6, 9, 9, 6, 9, 11, 7, 7, 9, 9, 5, 2, 8, 0, 8, 8, 9, 9, 9, 9, 3, 6, 2, 9, 9, 9, 9, 9, 9, 5, 9, 2, 4, 10, 10, 9, 9, 9, 9, 9, 9, 11, 7, 6, 11, 11, 9, 9, 9, 9, 9, 9, 5, 1, 4, 8, 12, 12

Offset: 0

N. J. A. Sloane, May 30 2025

The sixth and later columns consist of all 9's, and so the antidiagonals beyond that point also consist of all 9's.

			The first few antidiagonals are:
  1,
  2, 2,
  3, 3, 3,
  5, 5, 4, 4,
  0, 0, 7, 5, 5,
  5, 5, 3, 9, 6, 6,
  9, 9, 6, 9, 11, 7, 7,
  9, 9, 5, 2, 8, 0, 8, 8,
  9, 9, 9, 9, 3, 6, 2, 9, 9,
  ...

J. Froemke and J. W. Grossman, A mod-n Ackermann function, or what's so special about 1969?, Amer. Math. Monthly, 100 (1993), 180-183. See Fig. 1.

Cf. A085119.

n=12;a[i_,j_]:=a[i,j]=If[i==0,Mod[j+1,13],If[j==0,a[i-1,1],a[i-1,a[i,j-1]]]]; Flatten@Table[Diagonal[Reverse@Table[a[i,j],{i,0,n},{j,0,n}],k-n-1],{k,n}] (* Giorgos Kalogeropoulos, May 31 2025 *)

More terms from Giorgos Kalogeropoulos, May 31 2025

cp's OEIS Frontend

A352196 a(n) = number of steps for the standard mod-n Ackermann function to stabilize to a set consisting of only one value, or -1 if it does not stabilize.

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