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 1-5 of 5 results.

A249105 Numbers that form a Pythagorean 5-tuple with their first three arithmetic derivatives.

Original entry on oeis.org

4, 27, 1808, 3125, 12204, 12707, 82377, 269827, 823543, 1412500, 7089739, 9534375, 46873785, 78192979, 372241436
Offset: 1

Views

Author

Paolo P. Lava, Oct 21 2014

Keywords

Examples

			First three arithmetic derivatives of 1808 are 3632, 7280, 17616 and sqrt(1808^2 + 3632^2 + 7280^2 + 17616^2) = 19488.

Crossrefs

A051674 is a subsequence.
Cf. A003415, A210503, A230543, A249106, A249107, A249110.

Programs

  • Maple
    with(numtheory);
Dr:=proc(w) local x,p; x:=w*add(op(2,p)/op(1,p),p=ifactors(w)[2]); end:
P:=proc(q,h) local a,b,k,n; for n from 2 to q do a:=n; b:=n^2;
for k from 1 to h do a:=Dr(a); b:=b+a^2; od; if type(sqrt(b),integer) then print(n);
fi; od; end: P(10^9,3);

Extensions

a(13) from Ray Chandler, Dec 23 2016
a(14) from Ray Chandler, Dec 24 2016
a(15) from Ray Chandler, Jan 08 2017

A249106 Numbers that form a Pythagorean 6-tuple with their first four arithmetic derivatives.

Original entry on oeis.org

19164, 129357, 14971875, 45316123, 434325391
Offset: 1

Views

Author

Paolo P. Lava, Oct 21 2014

Keywords

Examples

			First four arithmetic derivatives of 19164 are 25564, 31848, 58412, 61916 and sqrt(19164^2 + 25564^2 + 31848^2 + 58412^2 + 61916^2) = 96336.

Crossrefs

Cf. A003415, A210503, A230543, A249105, A249107, A249110.

Programs

  • Maple
    with(numtheory);
Dr:=proc(w) local x,p; x:=w*add(op(2,p)/op(1,p),p=ifactors(w)[2]); end:
P:=proc(q,h) local a,b,k,n; for n from 2 to q do a:=n; b:=n^2;
for k from 1 to h do a:=Dr(a); b:=b+a^2; od; if type(sqrt(b),integer) then print(n);
fi; od; end: P(10^9,4);

Extensions

a(4) from Ray Chandler, Dec 23 2016
a(5) from Ray Chandler, Jan 11 2017

A249107 Numbers that form a Pythagorean 7-tuple with their first five arithmetic derivatives.

Original entry on oeis.org

4031, 10823, 416959, 496939, 1354980, 9146115, 38949392, 44472866, 262908396, 380264131
Offset: 1

Views

Author

Paolo P. Lava, Oct 21 2014

Keywords

Comments

If we consider Pythagorean 8-tuple and 9-tuple there are no terms up to n = 10^8.

Examples

			First five arithmetic derivatives of 4031 are 168, 332, 336, 832, 2560 and sqrt(4031^2 + 168^2 + 332^2 + 336^2 + 832^2 + 2560^2) = 4873.

Crossrefs

Cf. A003415, A210503, A230543, A249105, A249106, A249110.

Programs

  • Maple
    with(numtheory);
Dr:=proc(w) local x,p; x:=w*add(op(2,p)/op(1,p),p=ifactors(w)[2]); end:
P:=proc(q,h) local a,b,k,n; for n from 2 to q do a:=n; b:=n^2;
for k from 1 to h do a:=Dr(a); b:=b+a^2; od; if type(sqrt(b),integer) then print(n);
fi; od; end: P(10^9,5);

Extensions

a(5)-a(6) from Ray Chandler, Dec 22 2016
a(7)-a(8) from Ray Chandler, Dec 23 2016
a(9) from Ray Chandler, Jan 02 2017
a(10) from Ray Chandler, Jan 08 2017

A249110 Numbers that form a Pythagorean 10-tuple with their first eight arithmetic derivatives.

Original entry on oeis.org

4, 27, 3125, 398747, 823543
Offset: 1

Views

Author

Paolo P. Lava, Oct 21 2014

Keywords

Examples

			First eight arithmetic derivatives of 398747 are 1692, 2856, 5812, 5816, 8732, 9116, 9500, 15700 and sqrt(398747^2 + 1692^2 + 2856^2 + 5812^2 + 5816^2 + 8732^2 + 9116^2 + 9500^2 + 15700^2) = 399467.

Crossrefs

A051674 is a subsequence. - Ray Chandler, Dec 22 2016
Cf. A003415, A051674, A210503, A230543, A249105, A249106, A249107.

Programs

  • Maple
    with(numtheory);
Dr:=proc(w) local x,p; x:=w*add(op(2,p)/op(1,p),p=ifactors(w)[2]); end:
P:=proc(q,h) local a,b,k,n; for n from 2 to q do a:=n; b:=n^2;
for k from 1 to h do a:=Dr(a); b:=b+a^2; od; if type(sqrt(b),integer) then print(n);
fi; od; end: P(10^9,8);

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 1-5 of 5 results.

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