A228286 Smallest x + y*z, given x*y + z = n (for positive integers x, y, z).

2, 3, 4, 4, 6, 5, 7, 6, 6, 7, 9, 7, 10, 9, 8, 8, 12, 9, 12, 9, 10, 13, 15, 10, 10, 15, 12, 11, 15, 11, 16, 12, 14, 17, 12, 12, 17, 21, 16, 13, 18, 13, 19, 15, 14, 19, 24, 14, 14, 15, 20, 17, 21, 15, 16, 15, 22, 25, 28, 16, 22, 27, 16, 16, 18, 17, 23, 21, 25, 17

Offset: 2

Author

Andy Niedermaier, Aug 19 2013

Keywords

• nonn
• look

Comments

One of the terms in this sequence is the subject of a problem on the 2013 ARML contest (Team Round).

The Mathematica code below computes the quadruple {x, y, z, a(n)}, where z is as small as possible (in the event of a tie).

Examples

			For n = 160, a(n) = 50, as 26 * 6 + 4 = 160 and 26 + 6 * 4 = 50 and no triple of positive integers (x, y, z) with xy + z = 160 gives a smaller value for x + yz.
		

Links

• Peter Kagey, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A228287 (z-coordinate of the triple (x, y, z) that minimizes x + yz).

Cf. A228288 (least k such that z = n, given xy + z = k and x + yz is minimized).

Programs

Maple

A228286 := proc(n)
    local a,x,y,z ;
    a := n+n^2 ;
    for z from 1 to n-1 do
        for x in numtheory[divisors](n-z) do
            y := (n-z)/x ;
            a := min(a, x+y*z) ;
        end do:
    end do:
    return a;
end proc: # R. J. Mathar, Sep 02 2013

Mathematica

a[n_] := Module[{X, bX, bT, m},
  bT = n + 1;
  bX = {n - 1, 1, 1, n};
  X = bX;
  m = Floor[2*Sqrt[X[[3]]*(n - X[[3]])]];
  While[bT >= m && X[[3]] <= n/2,
   X[[2]] = Max[1, Floor[(n - bX[[3]])/bT]];
   While[X[[2]] <= Floor[bT/X[[3]]],
    If[Mod[n - X[[3]], X[[2]]] == 0,
     X[[1]] = (n - X[[3]])/X[[2]];
     X[[4]] = X[[1]] + X[[2]]*X[[3]];
     If[X[[4]] < bX[[4]], bX = X]];
    X[[2]] = X[[2]] + 1];
   X[[3]] = X[[3]] + 1;
   m = Floor[2*Sqrt[X[[3]]*(n - X[[3]])]]];
  Return[bX]]; Table[a[n][[-1]], {n, 2, 100}]