%I #15 Nov 18 2020 12:24:57

%S 42,84,168,252,672,756,420,504,2592,1872,840,1008,1512,2940,1680,2016,

%T 1260,4536,3360,3024,9450,4620,5880,6552,9504,6930,3780,8400,23184,

%U 25704,2520,6300,31752,8820,19800,11088,10920,13104,15840,19152,19656,16632,38016

%N Least k whose set of divisors contains exactly n Pythagorean quadruples, or 0 if no such k exists.

%C a(n) == 0 (mod 6).

%H Chai Wah Wu, <a href="/A331365/b331365.txt">Table of n, a(n) for n = 1..1026</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Pythagorean Quadruple.html">Pythagorean Quadruples</a>.

%e a(3) = 168 because the set of the divisors {1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168} contains 3 Pythagorean quadruples {2, 3, 6, 7}, {4, 6, 12, 14} and {8, 12, 24, 28}.

%p with(numtheory):

%p for n from 1 to 52 do :

%p ii:=0:

%p for q from 3 to 10^8 while(ii=0) do:

%p d:=divisors(q):n0:=nops(d):it:=0:

%p for i from 1 to n0-3 do:

%p for j from i+1 to n0-2 do :

%p for k from j+1 to n0-1 do:

%p for l from k+1 to n0 do:

%p if d[i]^2 + d[j]^2 + d[k]^2 = d[l]^2

%p then

%p it:=it+1:

%p else

%p fi:

%p od:

%p od:

%p od:

%p od:

%p if it = n

%p then

%p ii:=1: printf(`%d %d \n`,n,q):

%p else

%p fi:

%p od:

%p od:

%t upto = 38016; nq[n_] := If[Mod[n, 6] > 0, 0, Block[{t, u, v, c=0, d = Divisors@ n, m}, m = Length@ d; Do[t = d[[i]]^2 + d[[j]]^2; Do[u = t + d[[h]]^2; If[u > n^2, Break[]]; If[Mod[n^2, u] == 0 && IntegerQ[v = Sqrt@ u] && Mod[n, v] == 0, c++], {h, j+1, m-1}], {i, m-3}, {j, i+1, m-2}]; c]]; w = ParallelTable[ {nq@ n, n}, {n, 6 Range[ upto / 6]}]; t=0 Range@ Max[First /@ w]; Do[{q, x} = e; If[q > 0 && t[[q]] == 0, t[[q]] = x], {e, w}]; AppendTo[t, 0]; TakeWhile[t, # > 0 &] (* _Giovanni Resta_, May 04 2020 *)

%Y Cf. A027750, A169580, A330893, A330894.

%K nonn

%O 1,1

%A _Michel Lagneau_, May 03 2020