A096908 -id:A096908 - cp's OEIS frontend

A096907 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

1, 2, 4, 1, 6, 2, 3, 2, 2, 8, 1, 6, 6, 1, 8, 4, 4, 4, 6, 3, 3, 12, 9, 7, 10, 2, 2, 2, 12, 11, 3, 14, 6, 6, 5, 17, 8, 7, 4, 8, 4, 1, 15, 6, 1, 6, 8, 12, 3, 3, 19, 14, 2, 13, 2, 10, 23, 12, 9, 4, 4, 18, 7, 9, 2, 6, 2, 20, 4, 16, 13, 8, 5, 18, 18, 6, 11, 2, 6, 23, 12, 9, 4, 15, 12, 4, 22, 14, 1, 14

Offset: 1

Ray Chandler, Aug 15 2004

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Cf. A096908, A096909, A096910.

mx = 50; res = {}; Do[If[GCD[b, c, d] > 1, Continue[]]; If[IntegerQ[a = Sqrt[d^2 - b^2 - c^2]] && a > 0 && a <= b, AppendTo[res, {a, b, c, d}]], {d, mx}, {c, d}, {b, c}]; res[[All, 1]] (* Ivan Neretin, May 24 2015 *)

A096909 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

Original entry on oeis.org

2, 6, 7, 8, 7, 9, 12, 11, 14, 12, 12, 15, 17, 18, 16, 16, 19, 20, 18, 18, 22, 16, 20, 22, 23, 23, 25, 26, 21, 24, 24, 21, 22, 27, 30, 20, 25, 28, 28, 31, 32, 32, 26, 30, 30, 33, 27, 28, 28, 36, 26, 29, 29, 34, 34, 35, 24, 31, 32, 33, 39, 30, 30, 38, 39, 42, 42, 29, 35, 37, 40

Offset: 1

Ray Chandler, Aug 15 2004

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Cf. A096907, A096908, A096910.

mx = 50; res = {}; Do[If[GCD[b, c, d] > 1, Continue[]]; If[IntegerQ[a = Sqrt[d^2 - b^2 - c^2]] && a > 0 && a <= b, AppendTo[res, {a, b, c, d}]], {d, mx}, {c, d}, {b, c}]; res[[All, 3]] (* Ivan Neretin, May 24 2015 *)

A096910 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

Original entry on oeis.org

3, 7, 9, 9, 11, 11, 13, 15, 15, 17, 17, 19, 19, 19, 21, 21, 21, 21, 23, 23, 23, 25, 25, 27, 27, 27, 27, 27, 29, 29, 29, 31, 31, 31, 31, 33, 33, 33, 33, 33, 33, 33, 35, 35, 35, 35, 37, 37, 37, 37, 39, 39, 39, 39, 39, 39, 41, 41, 41, 41, 41, 43, 43, 43, 43, 43, 43, 45, 45, 45

Offset: 1

Ray Chandler, Aug 15 2004

nonn

Sequence with repetitions removed is A005818. - Ivan Neretin, May 24 2015

Ivan Neretin, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Cf. A096907, A096908, A096909 (other components of the quadruple), A046086, A046087, A020882 (Pythagorean triples ordered in a similar way).

mx = 50; res = {}; Do[If[GCD[b, c, d] > 1, Continue[]]; If[IntegerQ[a = Sqrt[d^2 - b^2 - c^2]] && a > 0 && a <= b, AppendTo[res, {a, b, c, d}]], {d, mx}, {c, d}, {b, c}]; res[[All, 4]] (* Ivan Neretin, May 24 2015 *)

A097263 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

Original entry on oeis.org

1, 3, 7, 1, 7, 9, 3, 11, 5, 9, 1, 15, 17, 1, 11, 13, 19, 5, 13, 3, 3, 15, 9, 7, 23, 23, 25, 7, 21, 11, 3, 21, 21, 27, 5, 17, 25, 7, 17, 31, 7, 1, 15, 17, 1, 33, 27, 21, 3, 3, 19, 29, 29, 13, 19, 35, 23, 31, 9, 33, 39, 25, 7, 9, 39, 7, 9, 29, 35, 37, 13, 19, 5, 27, 21, 27, 11, 21, 43, 23

Offset: 1

Ray Chandler, Aug 15 2004

nonn

One of a,b,c is odd, other two are even for primitive Pythagorean quadruples.

Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Cf. A096907, A096908, A096909.

A385356 Numbers x such that there exist two integers 00 such that sigma(x)^2 = sigma(y)^2 = x^2 + y^2 + z^2.

Original entry on oeis.org

2, 40, 164, 196, 224, 1120, 3040, 13440, 22932, 44200, 76160, 90848, 91720, 174592, 530200, 619840, 687184, 872960, 1686400, 1767040, 1807120, 1927680, 1990912, 2154880, 3653760, 4286880, 5637632, 5759680, 6442128, 8225280, 8943800, 9264320, 9465600, 9694080

Offset: 1

S. I. Dimitrov, Jun 26 2025

nonn

The numbers x, y and z form a sigma-quadratic triple. See Dimitrov link.

			(40, 58, 56) is such a triple because sigma(40)^2 = sigma(58)^2 = 90^2 = 40^2 + 58^2 + 56^2.

Chai Wah Wu, Table of n, a(n) for n = 1..161
S. I. Dimitrov, Generalizations of amicable numbers, arXiv:2408.07387 [math.NT], 2024.

Cf. A000203, A066784, A096907, A096908, A096909, A096910, A385325.

from itertools import count, islice
from sympy import divisor_sigma
from sympy.ntheory.primetest import is_square
def A385356_gen(startvalue=1): # generator of terms >= startvalue
    for x in count(max(startvalue,1)):
        sx, x2 = int(divisor_sigma(x)), x**2
        sx2 = sx**2
        if sx2>x2:
            for y in count(x):
                if (k:=sx2-x2-y**2)<=0:
                    break
                if is_square(k) and sx==divisor_sigma(y):
                    yield x
                    break
A385356_list = list(islice(A385356_gen(),8)) # Chai Wah Wu, Jul 02 2025

Data corrected by David A. Corneth, Jun 27 2025

a(18)-a(34) from Chai Wah Wu, Jul 02 2025

A097265 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

Original entry on oeis.org

2, 6, 4, 8, 6, 6, 12, 10, 14, 12, 12, 10, 6, 18, 16, 16, 8, 20, 18, 18, 22, 16, 20, 22, 10, 14, 10, 26, 16, 24, 24, 18, 22, 14, 30, 20, 20, 28, 28, 8, 32, 32, 26, 30, 30, 10, 24, 28, 28, 36, 26, 22, 26, 34, 34, 14, 24, 24, 32, 24, 12, 30, 30, 38, 18, 42, 42, 28, 28, 20, 40, 40

Offset: 1

Ray Chandler, Aug 15 2004

nonn

One of a,b,c is odd, other two are even for primitive Pythagorean quadruples.

Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Cf. A096907, A096908, A096909.

A097264 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

Original entry on oeis.org

2, 2, 4, 4, 6, 2, 4, 2, 2, 8, 12, 6, 6, 6, 8, 4, 4, 4, 6, 14, 6, 12, 12, 14, 10, 2, 2, 2, 12, 12, 16, 14, 6, 6, 6, 20, 8, 16, 4, 8, 4, 8, 18, 6, 18, 6, 8, 12, 24, 8, 22, 14, 2, 14, 2, 10, 24, 12, 24, 4, 4, 18, 30, 18, 2, 6, 2, 20, 4, 16, 16, 8, 8, 18, 18, 6, 18, 2, 6, 24, 12, 32, 4, 24, 12

Offset: 1

Ray Chandler, Aug 15 2004

nonn

One of a,b,c is odd, other two are even for primitive Pythagorean quadruples.

Eric Weisstein's World of Mathematics, Pythagorean Quadruple.

Cf. A096907, A096908, A096909.

A360946 Number of Pythagorean quadruples with inradius n.

Original entry on oeis.org

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

Offset: 1

Miguel-Ángel Pérez García-Ortega, Feb 26 2023

nonn

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.

			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).

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

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

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

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

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]]

A375098 Diagonals of a Euclidian solid such that there exists a Pythagorean quadruple d^2=a^2+b^2+c^2 that is more cube-like than any prior value of d.

Original entry on oeis.org

3, 9, 11, 41, 123, 153, 571, 1713, 2131, 7953, 23859, 29681, 110771, 332313, 413403, 1542841, 4628523, 5757961, 21489003, 64467009, 80198051, 299303201, 897909603, 1117014753, 4168755811, 12506267433, 15558008491, 58063278153, 174189834459, 216695104121

Offset: 1

Christian N. K. Anderson, Mark K. Transtrum, and David D. Allred, Jul 29 2024

nonn

To determine how "cube-like" a Pythagorean quad is, we use the quotient C/(C-a*b*c) where C is the volume of an ideal cube for a given diagonal d, C=(d/sqrt(3))^3. A ratio of 100 indicates that the best {a,b,c} combination creates a solid 1 part per 100 smaller than the ideal cube volume.
Contains all the terms in A001835 and A079935 except the leading 1s. For such a term b(m) contained in a(n) from those sequences, 2*b(m) will tie the current record, while 3*b(m) will tie but will also break the current record exactly half the time and thus appear as a(n+1). This means round((2+sqrt(3))*b(m)) will also be in the sequence as either a(n+1) or a(n+2) if a better quad for 3*b(n) was found.
The constant (2+sqrt(3)) follows from the recurrence relationship b(n)=4*b(n-1)-b(n-2). The constant can be represented as the continued fraction 3,1,2,1,2,1,2,1,2,.... The equivalents for the 2D case (A001653) are 3+2*sqrt(2) and {5,1,4,1,4,1,4,1,4,...}.
We conjecture this method provides complete solutions. This was confirmed directly by brute-force testing up to 1e7 (optimized using the sum-of-two-squares theorem for a given d^2-a^2 = b^2 + c^2; see link below).

			3 is in the sequence because 3^2=1^2+2^2+2^2 is the smallest Pythagorean quad, with an error of one part in 4.344.
6 is NOT in the sequence because {6,2,4,4} is the most cube-like Pythagorean quad, but only ties the previous record without breaking it.
7 is NOT in the sequence because the most cube-like quad {7,2,3,6} has an error of one part in 2.2, worse than that for d=3.
9 is in the sequence NOT because of {9,3,6,6} which ties the previous record, but because {9,4,4,7} improves on the previous record with an error of one part in 4.958.

Christian N. K. Anderson, Table of n, a(n) for n = 1..1000
Christian N. K. Anderson, Table of most cube-like Pythagorean quadruples and the corresponding and the error quotient for d=1..10000
Christian N. K. Anderson, Table of d, a, b, c and error quotient for n = 1..50.
Christian N. K. Anderson, Mathematica code for generating this sequence with different amounts of rigorous error checking.

Cf. A096907, A096908, A096909, A096910 for lists of a, b, c, and d of the 10,000 first Pythgorean quads, sorted by ascending d.
Contains all terms in A001835 and A079935 except the leading 1s.
Cf. A001653: The 2D equivalent of this sequence (i.e., right triangle whose legs are closest to equal)

(* An efficient program is provided in the links section. *)

For n == 0 (mod 3), a(n) = 4*a(n-2)-a(n-3) OR a(n) = floor(a(n-1)*(2+sqrt(3))/3),
For n == 1 (mod 3), a(n) = 4*a(n-2)-a(n-1) OR a(n) = floor(a(n-1)*(2+sqrt(3))),
For n == 2 (mod 3), a(n) = 3*a(n-1).

cp's OEIS Frontend

A096907 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

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A096909 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

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A096910 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

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A097263 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

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A385356 Numbers x such that there exist two integers 00 such that sigma(x)^2 = sigma(y)^2 = x^2 + y^2 + z^2.

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A097265 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

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A097264 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

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A360946 Number of Pythagorean quadruples with inradius n.

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A375098 Diagonals of a Euclidian solid such that there exists a Pythagorean quadruple d^2=a^2+b^2+c^2 that is more cube-like than any prior value of d.

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