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

2,1

Comments

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

Links

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

Example

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].

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 *)

Prog

(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), ", "))

Crossrefs

Cf. A073101, A192787, A292581.

Sequence in context: A110042 A306101 A364967 * A123027 A100652 A094416

Adjacent sequences: A337429 A337430 A337431 * A337433 A337434 A337435

Keyword

nonn,look

Author

Hugo Pfoertner, Oct 13 2020

Status

approved