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 1-4 of 4 results.

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

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 */

A097267 Least positive integer that is the diagonal of n primitive Pythagorean quadruples, or zero if no such integer exists.

Original entry on oeis.org

1, 3, 9, 19, 21, 27, 39, 33, 59, 51, 75, 57, 63, 103, 81, 0, 93, 121, 111, 99, 133, 123, 181, 129, 141, 153, 159, 211, 147, 227, 183, 171, 217, 263, 271, 201, 189, 219, 0, 307, 237, 243, 261, 249, 301, 267, 367, 485, 231, 291, 303, 409, 309, 419, 327, 297
Offset: 0

Views

Author

Ray Chandler, Aug 19 2004

Keywords

Comments

First appearance of n in A097266.
Conjecture: a(15)=a(38)=0.

Links

Crossrefs

Cf. A096907-A096909, A097263-A097266.

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

Views

Author

Miguel-Ángel Pérez García-Ortega, Feb 26 2023

Keywords

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.

Examples

			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).

References

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

Links

Crossrefs

Cf. A096907, A096908, A096909, A096910, A097263, A097264, A097265, A097266, A097267
Cf. A169580, A225206, A225207, A230543, A330893, A330894, A331365, A335654, A359741

Programs

  • 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]]
Showing 1-4 of 4 results.

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