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

2,1

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).

Links

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

Example

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.

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}]

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).

Sequence in context: A301764 A181833 A202650 * A158973 A071323 A071324

Adjacent sequences: A228283 A228284 A228285 * A228287 A228288 A228289

Keyword

nonn,look

Author

Andy Niedermaier, Aug 19 2013

Status

approved