A357554 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, then that is T(n,k).

1, 1, 2, 1, 2, 3, 1, 2, 2, 4, 1, 3, 3, 3, 5, 1, 2, 3, 3, 2, 6, 1, 4, 3, 4, 3, 4, 7, 1, 2, 4, 4, 4, 4, 2, 8, 1, 5, 3, 3, 5, 3, 3, 5, 9, 1, 2, 4, 4, 5, 5, 4, 4, 2, 10, 1, 6, 3, 5, 5, 6, 5, 5, 3, 6, 11, 1, 2, 3, 4, 4, 6, 6, 4, 4, 3, 2, 12, 1, 7, 5, 4, 5, 5, 7, 5, 5, 4, 5, 7, 13

Offset: 1

Author

J. M. Bergot and Robert Israel, Oct 02 2022

Keywords

• nonn
• tabl
• look

Examples

			For T(13,2) we have 2 -> floor(13/2) + (13 mod 2) = 7 -> floor(13/7) + (13 mod 7) = 7 so T(13,2) = 7.
Triangle starts:
  1;
  1, 2;
  1, 2, 3;
  1, 2, 2, 4;
  1, 3, 3, 3, 5;
  1, 2, 3, 3, 2, 6;
  1, 4, 3, 4, 3, 4, 7;
  1, 2, 4, 4, 4, 4, 2, 8;
  1, 5, 3, 3, 5, 3, 3, 5, 9;
  1, 2, 4, 4, 5, 5, 4, 4, 2, 10;
  ...
		

Links

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

Crossrefs

Cf. A357610.

Programs

Maple

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

Mathematica

T[n_, k_] := Module[{x, S}, S = {k}; x = k; While[True, x = Quotient[n, x] + Mod[n, x]; If[MemberQ[S, x], Return[x]]; S = S~Union~{x}]];
Table[T[n, k], {n, 1, 20}, {k, 1, n}] // Flatten (* Jean-François Alcover, Oct 16 2022, after Robert Israel *)

Python

def T(n, k):
    seen, x = set(), k
    while x not in seen: seen.add(x); q, r = divmod(n, x); x = q + r
    return x
print([T(n, k) for n in range(1, 14) for k in range(1, n+1)]) # Michael S. Branicky, Oct 04 2022

Formula

If k divides n, or if k > sqrt(n) and k^2-n is divisible by k-1, then T(n,k) = k.

T(n,2) = 2 if n is even, (n+1)/2 if n is odd.