A330893 Numbers whose set of divisors contains a Pythagorean quadruple.

42, 72, 84, 126, 144, 156, 168, 198, 210, 216, 252, 288, 294, 312, 330, 336, 342, 360, 378, 396, 420, 432, 462, 468, 504, 546, 570, 576, 588, 594, 624, 630, 648, 660, 672, 684, 714, 720, 756, 780, 792, 798, 840, 864, 882, 900, 924, 930, 936, 966, 990, 1008, 1026

Offset

1,1

Comments

A Pythagorean quadruple (x, y, z, m) is a set of positive integers that satisfy x^2 + y^2 + z^2 = m^2.

The corresponding number of quadruples of the sequence is 1, 1, 2, 2, 2, 1, 3, 1, 3, 2, 4, 3, 2, 2, 1, 4, 1, 2, 3, 2, 7, 4, ... (see the sequence A330894).

It is interesting to note that each set of divisors of a(n) contains m primitive Pythagorean quadruples for some n, m = 1, 2,...

Examples:

- The set of divisors of a(1)= 42 contains only one primitive Pythagorean quadruple: (2, 3, 6, 7).

- The set of divisors of a(9) = 210 contains two primitive Pythagorean quadruples: (2, 3, 6, 7) and (2, 5, 14, 15).

- The set of divisors of a(21) = 420 contains three primitive Pythagorean quadruples: (2, 3, 6, 7), (2, 5, 14, 15) and (4, 5, 20, 21).

If k is in the sequence then so is m*k for m > 1.

Assumes the elements (x,y,z,m) in a quadruple are distinct divisors, as otherwise 6 would be in the sequence with 1^2+2^2+2^2=3^2. - Chai Wah Wu, Nov 16 2020

Links

Table of n, a(n) for n=1..53.

Eric Weisstein's World of Mathematics, Pythagorean Quadruples.

Formula

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

Example

168 is in the sequence because the set of divisors  {1, 2, 3, 4, 6, 7, 8, 12, 14, 21, 24, 28, 42, 56, 84, 168} contains the Pythagorean quadruples {2, 3, 6, 7}, {4, 6, 12, 14} and {8, 12, 24, 28}. The first quadruple is primitive.

Maple

with(numtheory):
for n from 3 to 1200 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]^2 + d[j]^2 + d[k]^2 = d[m]^2
        then
        it:=it+1:
        else
       fi:
      od:
     od:
    od:
    od:
    if it>0 then
    printf(`%d, `, n):
    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]]^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]]; Select[ Range@ 1026, nq[#] > 0 &] (* Giovanni Resta, May 04 2020 *)

Prog

(PARI) isok(n) = {my(d=divisors(n), x); for (i=1, #d-3, for (j=i+1, #d-2, for (k=j+1, #d-1, if (issquare(d[i]^2 + d[j]^2 + d[k]^2, &x) && !(n % x), return(1)); ); ); ); } \\ Michel Marcus, Nov 16 2020

Crossrefs

Cf. A027750, A169580, A330894, A331365.

Sequence in context: A226168 A248430 A340384 * A305153 A340570 A261621

Adjacent sequences: A330890 A330891 A330892 * A330894 A330895 A330896

Keyword

nonn

Author

Michel Lagneau, May 01 2020

Status

approved