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%I #17 Jun 24 2020 04:58:47
%S 1,1,1,2,1,2,1,2,3,2,1,3,2,2,3,4,2,3,2,4,3,2,2,4,5,2,3,4,3,5,3,4,3,3,
%T 5,6,3,3,3,5,4,6,3,4,5,4,3,6,7,5,4,4,4,6,5,7,4,4,4,6,5,4,7,8,5,6,5,4,
%U 4,7,5,8,5,5,5,5,7,6,5,8,9,5,5,7,6,5,5,8,5,9,7,6,5,5,5,8,6,7,9,10,5,6,6,8
%N If n=sum a_i b_i, (a_i,b_i positive integers) then a(n)=max value of min(all a_i and b_i).
%C From _Robert Israel_, Aug 29 2018: (Start)
%C a(n) <= sqrt(n), with equality if n is a square.
%C a(n) >= A033676(n).
%C a(m+n) >= min(a(m), a(n)). (End)
%H Robert Israel, <a href="/A046805/b046805.txt">Table of n, a(n) for n = 1..10000</a>
%e a(13)=2 since 13=2*2+3*3.
%p A046805 := proc(n)
%p local p,a,abmin,divmin;
%p a := 0 ;
%p for p in combinat[partition](n) do
%p abmin := 1+n ;
%p for abprod in p do
%p divmin := A033676(abprod) ;
%p abmin := min(abmin,divmin) ;
%p end do:
%p a := max(a,abmin) ;
%p end do:
%p a ;
%p end proc: # _R. J. Mathar_, Oct 12 2015
%p # alternative program:
%p f:= proc(n) option remember; local v,a,b,vmax;
%p if issqr(n) then return sqrt(n) fi;
%p vmax:= 1;
%p for a from floor(sqrt(n)) by -1 while a > vmax do
%p for b from a to n/a do
%p v:= min(a, procname(n - a*b));
%p vmax:= max(vmax,v);
%p od od;
%p vmax
%p end proc:
%p f(0):= infinity:
%p map(f, [$1..200]); # _Robert Israel_, Aug 29 2018
%t f[n_] := f[n] = Module[{v, a, b, vMax}, If[IntegerQ[Sqrt[n]], Return[ Sqrt[n]]]; vMax = 1; For[a = Floor[Sqrt[n]], a > vMax, a--, For[b = a, b <= n/a, b++, v = Min[a, f[n - a b]]; vMax = Max[vMax, v]]]; vMax];
%t f[0] = Infinity;
%t Array[f, 200] (* _Jean-François Alcover_, Jun 23 2020, after _Robert Israel_ *)
%Y Cf. A033676.
%K easy,nice,nonn,look
%O 1,4
%A _Erich Friedman_
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