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
AUTHOR
J. M. Bergot and Robert Israel, Oct 02 2022
STATUS
approved