A331365 -id:A331365

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A330893

Numbers whose set of divisors contains a Pythagorean quadruple.

+10
5

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

(list; graph; refs; listen; history; text; internal format)

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:

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.

KEYWORD

nonn

AUTHOR

Michel Lagneau, May 01 2020

STATUS

approved

A335654

Numbers m such that the elements of all Pythagorean quadruples belonging to the set of divisors are exactly their first k divisors for some k.

+10
3

504, 1008, 1512, 1872, 2016, 3024, 3528, 3744, 4032, 4536, 5616, 6048, 6552, 7056, 7488, 8064, 9072, 9576, 10584, 11232, 12096, 13104, 13608, 14112, 14976, 16128, 16848, 17784, 18144, 19152, 19656, 21168, 21672, 22464, 23688, 24192, 24336, 24696, 26208, 27216

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Members m in A330893 for which there exists a number k < tau(m) such that the elements of all Pythagorean quadruples included in the set of the divisors of m are the first k divisors of m.

Conjecture 1: a(n) == 0 (mod 72).

Conjecture 2: if the numbers m such that the elements of all Pythagorean quadruples contained in the set of divisors of m are exactly the first k divisors of m, then k = tau(m) - 4 or tau(m) - 5.

The corresponding k of the sequence are given by the sequence {b(n)} = {20, 26, 28, 25, 32, 36, 32, 31, 38, 36, 35, 44, 44, 41,...} and the sequence {c(n)} = {tau(a(n)) - b(n)} = {4, 4, 4, 5, 4, 4, 4, 5, 4, 4, 5, 4, 4, 4, 5, 4, 4, 4, 4, 5, 4, 4, 4,...}. We observe that c(n) = 4 or 5 (see the table in the link). For n = 1, 2,...,400, the statistic observed is 301 occurrences for the number 4 (75.25 %) and 99 occurrences for the number 5 (24.75 %). It is probable that Pr(4) tends to .75 and Pr(5) tends to .25 when n tends into infinity, where Pr(x) is the probability of the occurrence x.

Assumes the elements in the quadruple are distinct. Otherwise 6, 12, 18, 24, ... are also terms. For instance the divisors of 6 are 1,2,3,6 and 1^2 + 2^2 + 2^2 = 3^2. - Chai Wah Wu, Nov 16 2020

LINKS

Chai Wah Wu, Table of n, a(n) for n = 1..1000

Michel Lagneau, Table

EXAMPLE

504 is in the sequence because the divisors are {1, 2, 3, 4, 6, 7, 8, 9, 12, 14, 18, 21, 24, 28, 36, 42, 56, 63, 72, 84, 126, 168, 252, 504} and the elements of the 8 Pythagorean quadruples belonging to the set of divisors of 504: (1, 4, 8, 9), (2, 3, 6, 7), (4, 6, 12, 14), (6, 9, 18, 21), (7, 28, 56, 63), (8, 12, 24, 28), (12, 18, 36, 42) and (24, 36, 72, 84) are the first 20 divisors of 504 with 20 = tau(504) - 4 = 24 - 4.

MAPLE

with(numtheory):

for n from 6 by 6 to 20000 do :lst:={}:lst1:={}:

d:=divisors(n):n0:=nops(d):

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

lst:=lst union {d[i]} union {d[j]} union {d[k]} union {d[m]}:

else

fi:

od:

n1:=nops(lst):

for l from 1 to n1 do:

lst1:= lst1 union {d[l]}:

od:

if lst=lst1 and lst<>{}

then

printf(`%d, `, n):

else fi:

od:

CROSSREFS

Cf. A000005, A330893, A330894, A331365.

KEYWORD

nonn

AUTHOR

Michel Lagneau, Jun 16 2020

STATUS

approved

A337098

Least k whose set of divisors contains exactly n quadruples (x, y, z, w) such that x^3 + y^3 + z^3 = w^3, or 0 if no such k exists.

+10
1

60, 120, 240, 432, 960, 360, 3840, 1728, 2592, 720, 1800, 2520, 161700, 1440, 6840, 9000, 2160, 2880, 168300, 5040, 41472, 5760, 1520820, 4320, 7200, 11520, 119700, 10080, 682080, 10800, 8640, 14400, 27360, 12960, 373248, 20160, 61560, 17280, 28800, 55440, 171000, 21600

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,1

COMMENTS

Observation: a(n) == 0 (mod 12).

Listing primitive tuples (w, x, y, z) enables to compute for some m how many such tuples are in its divisors using the lcm of such tuples. - David A. Corneth, Sep 26 2020

REFERENCES

Y. Perelman, Solutions to x^3 + y^3 + z^3 = u^3, Mathematics can be Fun, pp. 316-9 Mir Moscow 1985.

LINKS

David A. Corneth, Table of n, a(n) for n = 1..504

Fred Richman, Sums of Three Cubes

EXAMPLE

a(3) = 240 because the set of the divisors {1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240} contains 3 quadruples {3, 4, 5, 6}, {6, 8, 10, 12} and {12, 16, 20, 24}. The first quadruple is primitive.

MAPLE

with(numtheory):divisors(240);

for n from 1 to 52 do :

ii:=0:

for q from 6 by 6 to 10^8 while(ii=0) do:

d:=divisors(q):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:

if it = n

then

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

else

fi:

od:

MATHEMATICA

With[{s = Array[Count[Subsets[Divisors[#], {4}]^3, _?(#1 + #2 + #3 == #4 & @@ # &)] &, 10^4]}, Rest@ Values[#][[1 ;; 1 + LengthWhile[Differences@ Keys@ #, # == 1 &] ]] &@ KeySort@ PositionIndex[s][[All, 1]]] (* Michael De Vlieger, Sep 18 2020 *)

PROG

(Python)

from itertools import combinations

from sympy import divisors

def A337098(n):

k = 1

while True:

if n == sum(1 for x in combinations((d**3 for d in divisors(k)), 4) if sum(x[:-1]) == x[-1]):

return k

k += 1 # Chai Wah Wu, Sep 25 2020

CROSSREFS

Cf. A027750, A095868, A095867, A096545, A096546, A328204, A328149, A331365.

KEYWORD

nonn,hard

AUTHOR

Michel Lagneau, Aug 15 2020

EXTENSIONS

a(13)-a(22) from Chai Wah Wu, Sep 25 2020

More terms from David A. Corneth, Sep 26 2020

STATUS

approved

A360946

Number of Pythagorean quadruples with inradius n.

+10
1

1, 3, 6, 10, 9, 19, 16, 25, 29, 27, 27, 56, 31, 51, 49, 61, 42, 91, 52, 71, 89, 86, 63, 142, 64, 95, 116, 132, 83, 153, 90, 144, 149, 133, 108, 238, 108, 162, 169, 171, 122, 284, 130, 219, 200, 196, 145, 340, 174, 201, 231, 239, 164, 364, 176, 314, 278, 256, 190, 399, 195, 281, 360, 330

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

A Pythagorean quadruple is a quadruple (a,b,c,d) of positive integers such that a^2 + b^2 + c^2 = d^2 with a <= b <= c. Its inradius is (a+b+c-d)/2, which is a positive integer.

For every positive integer n, there is at least one Pythagorean quadruple with inradius n.

REFERENCES

J. M. Blanco Casado, J. M. Sánchez Muñoz, and M. A. Pérez García-Ortega, El Libro de las Ternas Pitagóricas, Preprint 2023.

LINKS

Table of n, a(n) for n=1..64.

Miguel-Ángel Pérez García-Ortega, Pythagorean Quadruples (in Spanish).

Wikipedia, Pythagorean quadruple.

EXAMPLE

For n=1 the a(1)=1 solution is (1,2,2,3).

For n=2 the a(2)=3 solutions are (1,4,8,9), (2,3,6,7) and (2,4,4,6).

For n=3 the a(3)=6 solutions are (1,6,18,19), (2,5,14,15), (2,6,9,11), (3,4,12,13), (3,6,6,9) and (4,4,7,9).

MATHEMATICA

n=50;

div={}; suc={}; A={};

Do[A=Join[A, {Range[1, (1+1/Sqrt[3])q]}], {q, 1, n}];

Do[suc=Join[suc, {Length[div]}]; div={}; For [i=1, i<=Length[Extract[A, q]], i++, div=Join[div, Intersection[Divisors[q^2+(Extract[Extract[A, q], i]-q)^2], Range[2(Extract[Extract[A, q], i]-q), Sqrt[q^2+(Extract[Extract[A, q], i]-q)^2]]]]], {q, 1, n}]; suc=Rest[Join[suc, {Length[div]}]]; matriz={{"q", " ", "cuaternas"}}; For[j=1, j<=n, j++, matriz=Join[matriz, {{j, " ", Extract[suc, j]}}]]; MatrixForm[Transpose[matriz]]

CROSSREFS

Cf. A096907, A096908, A096909, A096910, A097263, A097264, A097265, A097266, A097267

Cf. A169580, A225206, A225207, A230543, A330893, A330894, A331365, A335654, A359741

KEYWORD

nonn

AUTHOR

Miguel-Ángel Pérez García-Ortega, Feb 26 2023

STATUS

approved

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