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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).
2
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
1,2
COMMENTS
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.
LINKS
Robert Israel, Table of n, a(n) for n = 1..10011(rows 1 to 141, flattened)
EXAMPLE
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;
MAPLE
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;
MATHEMATICA
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 *)
CROSSREFS
Sequence in context: A097026 A189225 A169988 * A213259 A067597 A139394
KEYWORD
nonn,tabl
AUTHOR
J. M. Bergot and Robert Israel, Oct 27 2022
STATUS
approved