A337432 a(n) is the least value of z such that 4/n = 1/x + 1/y + 1/z with 0 < x <= y <= z has at least one solution.

2, 3, 3, 10, 6, 14, 6, 9, 10, 33, 9, 52, 14, 12, 12, 102, 18, 57, 15, 21, 22, 138, 18, 50, 26, 27, 21, 232, 24, 248, 24, 33, 34, 30, 27, 370, 38, 39, 30, 164, 35, 258, 33, 36, 46, 329, 36, 98, 50, 51, 39, 742, 54, 44, 42, 57, 58, 885, 45, 549, 62, 56, 48, 60, 66, 603

Offset: 2

Author

Hugo Pfoertner, Oct 13 2020

Keywords

• nonn
• look

Comments

See A073101 and A192787 for the history of the problem, references, and links.

Examples

			a(6)=6 because it is the least denominator z in the A192787(6)=8 solutions
  [x, y, z]: [2, 7, 42], [2, 8, 24], [2, 9, 18], [2, 10, 15], [2, 12, 12],
  [3, 4, 12], [3, 6, 6], [4, 4, 6];
a(13)=52 because the minimum of z in the A192787(13)=4 solutions is 52:
  [4, 18, 468], [4, 20, 130], [4, 26, 52], [5, 10, 130].
		

Links

• Hugo Pfoertner, Table of n, a(n) for n = 2..10000 (first 576 terms from Robert Israel)

Crossrefs

Cf. A073101, A192787, A292581.

Programs

Maple

f:= proc(n) local z,x,y;
  for z from floor(n/4)+1 do
    for x from floor(n*z/(4*z-n))+1 to z do
      y:= n*x*z/(4*x*z-n*x-n*z);
      if y::posint and y >= x and y <= z then return z fi
  od od
end proc:
map(f, [$2..100]); # Robert Israel, Oct 14 2020

Mathematica

a[n_] := For[z = Floor[n/4] + 1, True, z++, For[x = Floor[n(z/(4z - n))] + 1, x <= z, x++, y = n x z/(4 x z - n x - n z); If[IntegerQ[y] && x <= y <= z, Print[z]; Return [z]]]];
a /@ Range[2, 100] (* Jean-François Alcover, Oct 23 2020, after Robert Israel *)

PARI

a337432(n)={my(target=4/n,a,b,c,m=oo);for(a=1\target+1,3\target,my(t=target-1/a);for(b=max(1\t+1,a),2\t,c=1/(t-1/b);if(denominator(c)==1,m=min(m,max(a,max(b,c))))));m};
for(k=2,67,print1(a337432(k),", "))