A357610 Start with x = 3 and repeat the map x -> floor(n/x) + (n mod x) until an x occurs that has already appeared, then that is a(n).

1, 2, 3, 2, 3, 3, 3, 4, 3, 4, 3, 3, 5, 6, 3, 6, 5, 3, 7, 5, 3, 8, 7, 3, 9, 6, 3, 10, 9, 3, 11, 8, 3, 12, 5, 3, 13, 10, 3, 14, 9, 3, 15, 11, 3, 16, 11, 3, 17, 5, 3, 18, 5, 3, 19, 12, 3, 20, 11, 3, 21, 14, 3, 22, 5, 3, 23, 8, 3, 24, 15, 3, 25, 14, 3, 26, 17, 3, 27, 5, 3, 28, 11, 3, 29, 18, 3, 30, 9

Offset: 1

Author

J. M. Bergot and Robert Israel, Oct 06 2022

Keywords

• nonn
• look

Comments

1, 2 and the third column of A357554.

Examples

			a(4) = 2 because we have 3 -> floor(4/3) + (4 mod 3) = 2 -> floor(4/2) + (4 mod 2) = 2.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A357554.

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:
seq(g(n,3), n=1..100);

Mathematica

T[n_, k_] := Module[{x, S}, S = {k}; x = k; While[True, x = Total@QuotientRemainder[n, x]; If[MemberQ[S, x], Return[x]]; S = S~Union~{x}]];
a[n_] := T[n, 3];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Oct 17 2022, after Maple code *)

Python

def a(n):
    seen, x = set(), 3
    while x not in seen: seen.add(x); q, r = divmod(n, x); x = q + r
    return x
print([a(n) for n in range(1, 90)]) # Michael S. Branicky, Oct 06 2022~

Formula

a(n) = 3 if n is divisible by 3.

a(3*k+1) = k+1.

a(15*k+5) = 5 for k >= 1.

a(15*k+11) = 3*k+3.

a(255*k+137) = 15*k+9 for k >= 1.