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a(n) is the least m such that A358052(m,k) = n for some k.
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%I #13 Mar 12 2024 13:37:09

%S 1,2,5,8,14,20,32,38,59,59,63,116,122,158,158,218,278,278,402,548,642,

%T 642,642,642,642,1062,1062,1668,2474,2690,2690,2690,2690,2690,3170,

%U 3170,3170,3170,3170,3170,3170,9260,9260,9260,9788,9788,11772,11942,11942,11942,11942,11942

%N a(n) is the least m such that A358052(m,k) = n for some k.

%C a(n) is the least m such that iteration of the map x -> floor(m/x) + (m mod x), starting at some k in [1,m], produces n distinct values before repeating.

%e a(4) = 8 because A358052(8,6) = 4 and this is the first appearance of 4 in A358052.

%e Thus the map x -> floor(8/x) + (8 mod x) starting at 6 produces 4 distinct values before repeating: 6 -> 3 -> 4 -> 2 -> 4.

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

%p S:= {k};

%p x:= k;

%p for count from 1 do

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

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

%p S:= S union {x};

%p od

%p end proc:

%p V:= Vector(50): count:= 0:

%p for n from 1 while count < 50 do

%p for k from 1 to n do

%p v:= f(n,k);

%p if v <= 50 and V[v] = 0 then

%p V[v]:= n; count:= count+1;

%p fi

%p od od:

%p convert(V,list);

%t 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}]];

%t V = Table[0, {vmax = 40}]; count = 0;

%t For[n = 1, count < vmax, n++, For[k = 1, k <= n, k++, v = f[n, k]; If[v <= vmax && V[[v]] == 0, Print[n]; V[[v]] = n; count++]]];

%t V (* _Jean-François Alcover_, Mar 12 2024, after Maple code *)

%Y Cf. A234575, A358052.

%K nonn

%O 1,2

%A _J. M. Bergot_ and _Robert Israel_, Oct 27 2022