A358052 Triangular array read by rows. For T(n,k) where 1 <= k <= n, start with x = k and repeat the map x -> floor(n/x) + (n mod x) until an x occurs that has already appeared. The number of applications of the map is T(n,k).

1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 3, 3, 2, 2, 2, 1, 1, 2, 3, 2, 2, 2, 3, 2, 3, 4, 3, 2, 2, 2, 1, 2, 1, 3, 2, 3, 2, 2, 2, 2, 1, 2, 3, 2, 3, 3, 2, 2, 2, 2, 3, 2, 1, 3, 4, 3, 3, 2, 2, 2, 2, 2, 3, 2, 3, 4, 3, 3, 3, 2, 2, 2, 2, 1, 1, 3, 1, 4, 2, 2, 3, 3, 2, 2, 2, 4, 3, 3, 3, 2, 3

Offset: 1

Author

J. M. Bergot and Robert Israel, Oct 27 2022

Keywords

• nonn
• tabl

Comments

T(n,k) = 1 if k^2 >= n > k and k-1 divides n-k.

If A234575(n,k) = j then either T(n,k) = T(n,j)+1 and A357554(n,k) = A357554(n,j), or T(n,k) = T(n,j), A357554(n,k) = k and A357554(n,j) = j.

Examples

			For T(13,2) we have 2 -> floor(13/2) + (13 mod 2) = 7 -> floor(13/7) + (13 mod 7) = 7.  That is two applications of the map so T(13,2) = 2.
Triangle starts:
  1;
  2, 2;
  2, 1, 2;
  2, 1, 2, 2;
  2, 2, 1, 3, 2;
  2, 2, 2, 3, 3, 2;
  2, 2, 1, 1, 2, 3, 2;
  2, 2, 3, 2, 3, 4, 3, 2;
  2, 2, 1, 2, 1, 3, 2, 3, 2;
  2, 2, 2, 1, 2, 3, 2, 3, 3, 2;
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10011(rows 1 to 141, flattened)

Crossrefs

Cf. A234575, A357554.

Programs

Maple

f:= proc(n, k) local x, S,count;
  S:= {k};
  x:= k;
  for count from 1 do
     x:= iquo(n, x) + irem(n, x);
     if member(x, S) then return count fi;
     S:= S union {x};
  od
end proc:
for n from 1 to 20 do seq(f(n, k), k=1..n) od;

Mathematica

f[n_, k_] := Module[{x, S, count}, S = {k}; x = k; For[count = 1, True, count++, x = Quotient[n, x] + Mod[n, x]; If[MemberQ[S, x], Return[count]]; S = S~Union~{x}]];
Table[f[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 29 2023, after Maple code *)