A357610 - OEIS

%I #15 Oct 17 2022 09:45:45

%S 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,

%T 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,

%U 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

%N 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).

%C 1, 2 and the third column of A357554.

%H Robert Israel, <a href="/A357610/b357610.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

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

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

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

%p g:= proc(n, k) local x, S;

%p   S:= {k};

%p   x:= k;

%p   do

%p      x:= iquo(n, x) + irem(n, x);

%p      if member(x, S) then return x fi;

%p      S:= S union {x};

%p   od

%p end proc:

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

%t 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}]];

%t a[n_] := T[n, 3];

%t Table[a[n], {n, 1, 100}] (* _Jean-François Alcover_, Oct 17 2022, after Maple code *)

%o (Python)

%o def a(n):

%o     seen, x = set(), 3

%o     while x not in seen: seen.add(x); q, r = divmod(n, x); x = q + r

%o     return x

%o print([a(n) for n in range(1, 90)]) # _Michael S. Branicky_, Oct 06 2022~

%Y Cf. A357554.

%K nonn,look

%O 1,2

%A _J. M. Bergot_ and _Robert Israel_, Oct 06 2022