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A328204 Numbers of quadruples contained in the divisors of A328149(n).

%I A328204 #62 Nov 15 2020 08:24:25
%S A328204 1,1,2,2,2,2,3,3,2,6,2,4,4,2,3,4,4,3,2,10,2,2,4,6,4,2,5,4,2,10,2,5,6,
%T A328204 2,4,6,4,2,2,14,3,4,4,4,4,2,6,1,8,2,11,2,4,6,4,4,6,4,2,4,17,2,2,4,6,4,
%U A328204 4,1,8,4,2,12,2,9,6,2,6,4,4,4,2,18,3,2,6
%N A328204 Numbers of quadruples contained in the divisors of A328149(n).
%C A328204 A quadruple (x, y, z, w) of A328149 is a set of positive integers that satisfy x^3 + y^3 + z^3 = w^3.
%H A328204 Michel Marcus, <a href="/A328204/b328204.txt">Table of n, a(n) for n = 1..10000</a>
%e A328204 a(7) = 3 because the set of divisors of A328149(7) = 240: {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240} contains the three quadruples {3, 4, 5, 6}, {6, 8, 10, 12} and {12, 16, 20, 24}. The first quadruple is primitive.
%p A328204 with(numtheory):
%p A328204 for n from 3 to 3000 do :
%p A328204    d:=divisors(n):n0:=nops(d):it:=0:
%p A328204     for i from 1 to n0-3 do:
%p A328204      for j from i+1 to n0-2 do :
%p A328204       for k from j+1 to n0-1 do:
%p A328204       for m from k+1 to n0 do:
%p A328204        if d[i]^3 + d[j]^3 + d[k]^3 = d[m]^3
%p A328204         then
%p A328204         it:=it+1:
%p A328204         else
%p A328204        fi:
%p A328204       od:
%p A328204      od:
%p A328204     od:
%p A328204     od:
%p A328204     if it>0 then
%p A328204     printf(`%d, `,it):
%p A328204     else fi:
%p A328204    od:
%t A328204 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]]^3 + d[[j]]^3; Do[u = t + d[[h]]^2; If[u > n^3, Break[]]; If[Mod[n^3, u] == 0 && IntegerQ[v = u^(1/3)] && Mod[n, v] == 0, c++], {h, j+1, m-1}], {i, m-3}, {j, i+1, m - 2}]; c]]; Select[Array[nq, 1638], # > 0 &] (* program from _Giovanni Resta_ adapted for the sequence. See A330894 *)
%o A328204 (PARI) isok(n) = {my(d=divisors(n), nb=0, m); if (#d > 3, for (i=1, #d-3, for (j=i+1, #d-2, for (k=j+1, #d-1, if (ispower(d[i]^3+d[j]^3+d[k]^3, 3, &m) && !(n%m), nb++););););); nb;}
%o A328204 lista(nn) = my(m); for (n=1, nn, if (m=isok(n), print1(m, ", "))); \\ _Michel Marcus_, Nov 15 2020
%Y A328204 Cf. A027750, A096545, A328149, A330894.
%K A328204 nonn
%O A328204 1,3
%A A328204 _Michel Lagneau_, Jun 07 2020