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

1,3

Links

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

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.

Example

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;
  ...

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 *)

Prog

(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

Crossrefs

Cf. A357610.

Sequence in context: A334230 A275723 A198338 * A199086 A098053 A272907

Adjacent sequences: A357551 A357552 A357553 * A357555 A357556 A357557

Keyword

nonn,tabl,look

Author

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

Status

approved