A228287 Smallest value of z in the minimal value of x + y*z, given x*y + z = n (where x, y, z are positive integers).

1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 2, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 2, 3, 1, 2, 3, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 2, 3, 1, 1, 1, 1, 2, 1, 1, 2, 3, 1, 1, 2, 1

Offset: 2

Author

Andy Niedermaier, Aug 19 2013

Keywords

• nonn

Comments

If there are multiple triples (x, y, z) for which xy + z = n and x + yz is minimized, consider the triple with smallest z. I.e., this sequence illustrates the smallest z needed to minimize x + y*z.

Examples

			For n = 215 the triples (53, 4, 3) and (35, 6, 5) both give the minimal value of x + yz = 65. Thus a(215) = 3.
		

Crossrefs

Cf. A228286.

Programs

Maple

A228287 := proc(n)
    local a,x,y,z,zfin ;
    a := n+n^2 ;
    zfin := n ;
    for z from 1 to n-1 do
        for x in numtheory[divisors](n-z) do
            y := (n-z)/x ;
            if x+y*z < a then
                a := x+y*z ;
                zfin := z ;
            end if;
        end do:
    end do:
    return zfin;
end proc: # R. J. Mathar, Sep 02 2013

Mathematica

A228287[n_] := Module[{a, x, y, z, zfin}, a = n + n^2; zfin = n; Do[Do[y = (n-z)/x; If[x + y*z < a, a = x + y*z; zfin = z], {x, Divisors[n-z]}], {z, 1, n-1}]; zfin];
Table[A228287[n], {n, 2, 100}] (* Jean-François Alcover, Aug 08 2023, after R. J. Mathar *)