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 10 results.

A106430 Ordered and uniqued length of side opposite the greater of the two angles, one being the double of the other, of a primitive integer-sided triangle.

Original entry on oeis.org

6, 12, 15, 20, 28, 30, 35, 40, 42, 45, 56, 63, 66, 70, 72, 77, 84, 88, 90, 91, 99, 104, 110, 117, 120, 126, 130, 132, 143, 144, 153, 154, 156, 165, 170, 176, 182, 187, 190, 195, 198, 204, 208, 209, 210, 220, 221, 228, 231, 234, 238, 240, 247, 255, 260, 266, 272, 273, 276, 285, 286, 299
Offset: 1

Views

Author

Lekraj Beedassy and Ray Chandler, May 09 2005

Keywords

Comments

This is also the list of a, where a is positive integer solutions of 1/a+1/b=1/c and a>b>c and gcd[a,b,c]=1, sorted by ascending a, then b. See A247372 for list of b, A246429 for list of c. - Albert Lau, Sep 19 2014

Links

Crossrefs

Cf. A106499-A106506, A106410, A106420, A247372, A246429

Programs

  • Mathematica
    aMax = 300;
Select[Sequence @@@ Table[{m (m + n), n (m + n), m n}, {m, Sqrt[aMax]}, {n, Min[m - 1, aMax/m - m]}], GCD @@ # == 1 &] // Sort;
%[[;; , 1]]
(* Albert Lau, Sep 19 2014 *)

Formula

Values r*s, where r
The other 2 sides are s^2 and r^2-s^2. - Albert Lau, Sep 19 2014

A106499 Ordered perimeter of primitive integer-sided triangle one of whose angles is twice another.

Original entry on oeis.org

15, 28, 40, 45, 66, 77, 84, 91, 104, 120, 126, 144, 153, 170, 187, 190, 198, 209, 220, 228, 231, 260, 273, 276, 286, 299, 312, 322, 325, 345, 350, 378, 390, 400, 420, 432, 435, 442, 459, 464, 476, 493, 496, 510, 522, 527, 544, 551, 558, 561, 570, 589, 608
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2005

Keywords

Comments

For corresponding values of r and s, see A106500 and A106501.
The sides of the primitive triangle are given by the triple (r*s,s^2 - r^2,r^2).

Links

Crossrefs

Cf. A106500-A106506, A106410, A106420, A106430.

Programs

  • Mathematica
    s[p_] := Solve[1 < r < s < 2r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers];
Reap[For[p = 1, p <= 10000, p++, sp = s[p]; If[sp =!= {}, Sow[p]; Print[p, " ", {r s, s^2 - r^2, r^2} /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 06 2020 *)

Formula

a(n)=s*(r+s), where r

Extensions

Extended by Ray Chandler, May 09 2005

A106420 Length of side opposite the greater of the two angles, one being the double of the other, of a primitive integer-sided triangle, sorted on perimeter (A106499).

Original entry on oeis.org

6, 12, 15, 20, 30, 28, 35, 42, 40, 56, 45, 63, 72, 70, 66, 90, 77, 88, 99, 84, 110, 91, 104, 132, 117, 130, 143, 126, 156, 120, 154, 182, 165, 144, 195, 176, 210, 153, 170, 208, 187, 204, 240, 221, 198, 238, 255, 190, 234, 272, 209, 228, 247, 220, 266, 306, 285
Offset: 1

Views

Author

Lekraj Beedassy and Ray Chandler, May 09 2005

Keywords

Crossrefs

Cf. A106499-A106506, A106410, A106430.

Programs

  • Mathematica
    sol[p_] := Solve[1 < r < s < 2 r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers];
Reap[For[p = 1, p <= 1000, p++, sp = sol[p]; If[sp =!= {}, Print[r s /. sp[[1]]]; Sow[r s /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)

Formula

a(n) = A106500(n) * A106501(n).

A106506 Length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle, sorted on perimeter (A106499).

Original entry on oeis.org

5, 7, 16, 9, 11, 33, 24, 13, 39, 15, 56, 32, 17, 51, 85, 19, 72, 57, 40, 95, 21, 120, 105, 23, 88, 69, 48, 115, 25, 161, 75, 27, 104, 175, 56, 135, 29, 208, 189, 87, 168, 145, 31, 120, 203, 93, 64, 261, 155, 33, 240, 217, 192, 279, 165, 35, 136, 231, 105, 320, 72, 37, 272
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2005

Keywords

Crossrefs

Cf. A106499-A106505, A106410, A106420, A106430.

Programs

  • Mathematica
    sol[p_] := Solve[1 < r < s < 2 r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers];
Reap[For[p = 1, p <= 1000, p++, sp = sol[p]; If[sp =!= {}, Print[s^2 - r^2 /. sp[[1]]]; Sow[s^2 - r^2 /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)

Formula

a(n) = A106501(n)^2 - A106500(n)^2.

Extensions

Extended by Ray Chandler, May 09 2005

A106500 Values r associated with A106499.

Original entry on oeis.org

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

Views

Author

Lekraj Beedassy, May 04 2005

Keywords

Crossrefs

Cf. A106499-A106506, A106410, A106420, A106430.

Programs

  • Mathematica
    s[p_] := Solve[1 < r < s < 2r && p == r s + s^2 && GCD[r, s] == 1, {r, s}, Integers];
Reap[For[p = 1, p <= 2000, p++, sp = s[p]; If[sp =!= {}, Print[r /. sp[[1]] ]; Sow[r /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 06 2020 *)

Extensions

Extended by Ray Chandler, May 09 2005

A106505 Ordered and uniqued length of side common to the two angles, one being the double of the other, of a primitive integer-sided triangle.

Original entry on oeis.org

5, 7, 9, 11, 13, 15, 16, 17, 19, 21, 23, 24, 25, 27, 29, 31, 32, 33, 35, 37, 39, 40, 41, 43, 45, 47, 48, 49, 51, 53, 55, 56, 57, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 75, 77, 79, 80, 81, 83, 85, 87, 88, 89, 91, 93, 95, 96, 97, 99, 101, 103, 104, 105, 107, 109, 111, 112
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2005

Keywords

Comments

The terms are proposed without repetition. For example, there exist two such triangles with a length of side = 33. They correspond respectively to s^2 - r^2 = 33 (see formula) with (r, s) = (4, 7) and sides (33, 28, 16), and the other triangle with (r, s) = (16, 17) and sides (33, 272, 256). Lengths = 39, 51, 57, 69, 75, ... correspond to two distinct triangles ... The lengths of these sides are proposed with repetition in A343064. - Bernard Schott, Apr 22 2021

Crossrefs

Cf. A106499-A106506, A106410, A106420, A106430, A321499.
Cf. A343063, A343064, A343065, A343066, A343067.

Formula

Values s^2 - r^2, where r
Conjecture: for n>2, a(n+5) = a(n) + 8. - Ralf Stephan, Nov 16 2010.
Empirical g.f.: -x*(x^7+x^6+3*x^5-2*x^4-2*x^3-2*x^2-2*x-5) / ((x-1)^2*(x^4+x^3+x^2+x+1)). - Colin Barker, Oct 05 2013

Extensions

Extended by Ray Chandler, May 09 2005

A106501 Values s associated with A106499.

Original entry on oeis.org

3, 4, 5, 5, 6, 7, 7, 7, 8, 8, 9, 9, 9, 10, 11, 10, 11, 11, 11, 12, 11, 13, 13, 12, 13, 13, 13, 14, 13, 15, 14, 14, 15, 16, 15, 16, 15, 17, 17, 16, 17, 17, 16, 17, 18, 17, 17, 19, 18, 17, 19, 19, 19, 20, 19, 18, 19, 20, 19, 21, 19, 19, 21, 20, 22, 21, 20, 21, 23, 22, 23, 21, 23, 22
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2005

Keywords

Crossrefs

Cf. A106499-A106506, A106410, A106420, A106430.

Extensions

Extended by Ray Chandler, May 09 2005

A106502 Shortest side of primitive integer-sided triangle having an angle twice another, sorted on perimeter (A106499).

Original entry on oeis.org

4, 7, 9, 9, 11, 16, 24, 13, 25, 15, 25, 32, 17, 49, 36, 19, 49, 57, 40, 49, 21, 49, 64, 23, 81, 69, 48, 81, 25, 64, 75, 27, 104, 81, 56, 121, 29, 81, 100, 87, 121, 144, 31, 120, 121, 93, 64, 100, 155, 33, 121, 144, 169, 121, 165, 35, 136, 169, 105, 121, 72, 37, 169, 111, 169
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2005

Keywords

Comments

Triple is (A106502,106503,106504).

Crossrefs

Cf. A106499-A106506, A106410, A106420, A106430.

Extensions

Corrected and extended by Ray Chandler, May 09 2005

A106503 Middle side of primitive integer-sided triangle having an angle twice another, sorted on perimeter (A106499).

Original entry on oeis.org

5, 9, 15, 16, 25, 28, 25, 36, 39, 49, 45, 49, 64, 51, 66, 81, 72, 64, 81, 84, 100, 91, 104, 121, 88, 100, 121, 115, 144, 120, 121, 169, 121, 144, 169, 135, 196, 153, 170, 169, 168, 145, 225, 169, 198, 196, 225, 190, 169, 256, 209, 217, 192, 220, 196, 289, 225, 231
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2005

Keywords

Comments

Triple is (A106502, A106503, A106504).

Crossrefs

Cf. A106499-A106506, A106410, A106420, A106430.

Extensions

Corrected and extended by Ray Chandler, May 09 2005

A106504 Longest side of primitive integer-sided triangle having an angle twice another, sorted on perimeter (A106499).

Original entry on oeis.org

6, 12, 16, 20, 30, 33, 35, 42, 40, 56, 56, 63, 72, 70, 85, 90, 77, 88, 99, 95, 110, 120, 105, 132, 117, 130, 143, 126, 156, 161, 154, 182, 165, 175, 195, 176, 210, 208, 189, 208, 187, 204, 240, 221, 203, 238, 255, 261, 234, 272, 240, 228, 247, 279, 266, 306, 285
Offset: 1

Views

Author

Lekraj Beedassy, May 04 2005

Keywords

Comments

Triple is (A106502, A106503, A106504).

Links

Crossrefs

Cf. A106499-A106506, A106410, A106420, A106430.

Extensions

Corrected and extended by Ray Chandler, May 09 2005
Showing 1-10 of 10 results.

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