cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-10 of 14 results. Next

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

Views

Author

Ray Chandler, Aug 15 2004

Keywords

Links

Crossrefs

Cf. A096907, A096908, A096910.

Programs

  • Mathematica
    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

Views

Author

Ray Chandler, Aug 15 2004

Keywords

Comments

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)

A096908 Primitive Pythagorean Quadruples a^2+b^2+c^2=d^2, 0

Original entry on oeis.org

2, 3, 4, 4, 6, 6, 4, 10, 5, 9, 12, 10, 6, 6, 11, 13, 8, 5, 13, 14, 6, 15, 12, 14, 10, 14, 10, 7, 16, 12, 16, 18, 21, 14, 6, 20, 20, 16, 17, 8, 7, 8, 18, 17, 18, 10, 24, 21, 24, 8, 22, 22, 26, 14, 19, 14, 24, 24, 24, 24, 12, 25, 30, 18, 18, 7, 9, 28, 28, 20, 16, 19, 8, 27, 21, 27, 18
Offset: 1

Views

Author

Ray Chandler, Aug 15 2004

Keywords

Links

Crossrefs

Cf. A096907, A096909, A096910.

Programs

  • Mathematica
    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, 2]] (* Ivan Neretin, May 24 2015 *)

A230543 Numbers n that form a Pythagorean quadruple with n', n'' and sqrt(n^2 + n'^2 + n''^2), where n' and n'' are the first and the second arithmetic derivative of n.

Original entry on oeis.org

512, 1203, 3456, 6336, 23328, 42768, 157464, 249753, 288684, 400000, 722718, 1062882, 1948617, 2700000, 4950000, 18225000, 33412500, 105413504, 123018750, 225534375, 312500000, 408918816
Offset: 1

Views

Author

Paolo P. Lava, Oct 25 2013

Keywords

Comments

Tested up to n = 4.09*10^8.

Examples

			If n = 6336 then n' = 23808, n'' = 103936 and sqrt(n^2 + n'^2 + n''^2) = 106816.

Links

Crossrefs

Cf. A003415, A009003, A009004, A068346, A210503, A211176.
Cf. A096907-A096909 and A097263-A097266 for Pythagorean Quadruples.

Programs

  • Maple
    with(numtheory): P:= proc(q) local a1, a2, n, p;
for n from 2 to q do a1:=n*add(op(2,p)/op(1,p),p=ifactors(n)[2]);
a2:=a1*add(op(2,p)/op(1,p),p=ifactors(a1)[2]);
if type(sqrt(n^2+a1^2+a2^2),integer) then print(n);
fi; od; end: P(10^10);

Extensions

a(16)-a(18) from Giovanni Resta, Oct 25 2013
a(19) from Ray Chandler, Dec 22 2016
a(20) from Ray Chandler, Dec 31 2016
a(21) from Ray Chandler, Jan 05 2017
a(22) from Ray Chandler, Jan 09 2017

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

Views

Author

Ray Chandler, Aug 15 2004

Keywords

Comments

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

Links

Crossrefs

Cf. A096907, A096908, A096909.

A097266 Number of primitive Pythagorean quadruples with diagonal 2n+1.

Original entry on oeis.org

0, 1, 0, 1, 2, 2, 1, 2, 2, 3, 4, 3, 2, 5, 3, 4, 7, 4, 4, 6, 5, 6, 6, 6, 7, 9, 6, 6, 11, 8, 7, 12, 5, 9, 12, 9, 9, 10, 12, 10, 14, 11, 7, 14, 11, 12, 16, 10, 12, 19, 12, 13, 16, 14, 13, 18, 14, 12, 18, 16, 17, 21, 12, 16, 23, 17, 20, 18, 17, 18, 24, 18, 13, 28, 18, 19, 25, 16, 19, 26, 24
Offset: 0

Views

Author

Ray Chandler, Aug 16 2004

Keywords

Comments

There are no such quadruples with diagonal 2n. - Michael Somos, Nov 17 2018

Links

Crossrefs

Cf. A096907-A096909, A097263-A097265.

Programs

  • Mathematica
    a[ n_] := With[ {w = 2 n + 1}, Sum[ Boole[x^2 + y^2 + z^2 == w^2 && 1 == GCD[x, y, z, w]], {z, w - 1}, {y, z}, {x, y}]]; (* Michael Somos, Nov 17 2018 *)
  • PARI
    {a(n) = my(w = 2*n+1); sum(z=1, w-1, sum(y=1, z, sum(x=1, y,  x^2 + y^2 + z^2 == w^2 && 1 == gcd([x, y, z, w]))))}; /* Michael Somos, Nov 17 2018 */

A225207 Number of primitive Pythagorean quadruples (a, b, c, d) with d < 10^n.

Original entry on oeis.org

4, 347, 34163, 3412143, 341175325, 34117055318, 3411700692939, 341170025540426, 34117002022924247
Offset: 1

Views

Author

Arkadiusz Wesolowski, May 01 2013

Keywords

Comments

a(n) ~ (1+G)*A225206(n)/Pi, where G is Catalan's constant (A006752).

Examples

			a(1) = 4 because there are four primitive solutions (a, b, c, d) as follows: (1, 2, 2, 3), (2, 3, 6, 7), (1, 4, 8, 9), (4, 4, 7, 9) with d < 10.

Links

Crossrefs

Cf. A096907-A096910, A225206.

Extensions

a(4) from Giovanni Resta, May 01 2013
a(5)-a(9) from Max Alekseyev, Feb 28 2023

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

Views

Author

S. I. Dimitrov, Jun 26 2025

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Python
    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

Extensions

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

Views

Author

Ray Chandler, Aug 15 2004

Keywords

Comments

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

Links

Crossrefs

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

Views

Author

Ray Chandler, Aug 15 2004

Keywords

Comments

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

Links

Crossrefs

Cf. A096907, A096908, A096909.
Showing 1-10 of 14 results. Next

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