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

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, 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, 4, 1, 8, 4, 2, 12, 2, 9, 6, 2, 6, 4, 4, 4, 2, 18, 3, 2, 6

Offset

1,3

Comments

A quadruple (x, y, z, w) of A328149 is a set of positive integers that satisfy x^3 + y^3 + z^3 = w^3.

Links

Michel Marcus, Table of n, a(n) for n = 1..10000

Example

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.

Maple

with(numtheory):
for n from 3 to 3000 do :
   d:=divisors(n):n0:=nops(d):it:=0:
    for i from 1 to n0-3 do:
     for j from i+1 to n0-2 do :
      for k from j+1 to n0-1 do:
      for m from k+1 to n0 do:
       if d[i]^3 + d[j]^3 + d[k]^3 = d[m]^3
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
    od:
    if it>0 then
    printf(`%d, `, it):
    else fi:
   od:

Mathematica

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 *)

Prog

(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; }
lista(nn) = my(m); for (n=1, nn, if (m=isok(n), print1(m, ", "))); \\ Michel Marcus, Nov 15 2020

Crossrefs

Cf. A027750, A096545, A328149, A330894.

Sequence in context: A224713 A357427 A168557 * A194320 A231555 A120425

Adjacent sequences: A328201 A328202 A328203 * A328205 A328206 A328207

Keyword

nonn

Author

Michel Lagneau, Jun 07 2020

Status

approved