A357554 -id:A357554 - cp's OEIS frontend

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

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

1, 2 and the third column of A357554.

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

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

Cf. A357554.

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);

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 *)

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~

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.

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).

Original entry on oeis.org

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

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

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.

			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;

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

Cf. A234575, A357554.

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;

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 *)

cp's OEIS Frontend

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).

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