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 11 results. Next

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

A106410 Length of side opposite the lesser 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

4, 9, 9, 16, 25, 16, 25, 36, 25, 49, 25, 49, 64, 49, 36, 81, 49, 64, 81, 49, 100, 49, 64, 121, 81, 100, 121, 81, 144, 64, 121, 169, 121, 81, 169, 121, 196, 81, 100, 169, 121, 144, 225, 169, 121, 196, 225, 100, 169, 256, 121, 144, 169, 121, 196, 289, 225, 169, 256, 121
Offset: 1

Views

Author

Lekraj Beedassy and Ray Chandler, May 09 2005

Keywords

Crossrefs

Cf. A106499-A106506, 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[r^2 /. sp[[1]]]; Sow[r^2 /. sp[[1]]]]]][[2, 1]] (* Jean-François Alcover, Mar 07 2020 *)

Formula

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

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

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

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

A343067 Perimeter of integer-sided primitive triangles (a, b, c) whose angle B = 2*C.

Original entry on oeis.org

15, 28, 45, 66, 91, 120, 40, 153, 190, 231, 276, 84, 325, 378, 435, 496, 144, 77, 561, 630, 703, 104, 780, 220, 861, 946, 1035, 1128, 312, 1225, 170, 1326, 1431, 1540, 126, 420, 209, 1653, 1770, 1891, 2016, 544, 2145, 2278, 299, 2415, 2556, 198, 684, 2701, 350, 2850, 3003, 3160
Offset: 1

Views

Author

Bernard Schott, Apr 15 2021

Keywords

Comments

The triples (a, b, c) are listed in increasing order of side a, and if sides a coincide then in increasing order of side b.
This sequence is nonincreasing: a(7) = 40 < a(6) = 120.
If in triangle ABC, B = 2*C, then the corresponding metric relation between sides is a*c + c^2 = c * (a + c) = b^2.
As the metric relation is equivalent to a = m^2 - k^2, b = m*k, c = k^2, with gcd(m,k) = 1 and k < m < 2k, so all terms are of the form m^2 + m*k = m * (m+k) with gcd(m,k) = 1 and k < m < 2k. These perimeters are in increasing order in A106499.
For the corresponding primitive triples and miscellaneous properties and references, see A343063.

Examples

			According to inequalities between a, b, c, there exist 3 types of such triangles:
a(1) = 15 with c < a < b for the first triple (5, 6, 4);
a(7) = 40 with c < b < a for the seventh triple (16, 15, 9);
a(8) = 153 with a < c < b for the eighth triple (17, 72, 64).

Crossrefs

Cf. A335897 (similar for A < B < C in arithmetic progression).
Cf. A343063 (triples), A343064 (side a), A343065 (side b), A343066 (side c), A106499 (perimeters in increasing order).

Programs

  • Maple
    for a from 2 to 100 do
for c from 3 to floor(a^2/2) do
d := c*(a+c);
if issqr(d) and igcd(a, sqrt(d), c)=1 and abs(a-c)
  • PARI
    lista(nn) = {for (a = 2, nn, for (c = 3, a^2\2, my(d = c*(a+c)); if (issquare(d) && (gcd([a, sqrtint(d), c])==1) && (abs(a-c)Michel Marcus, May 12 2022

Formula

a(n) = A343063(n, 1) + A343063(n, 2) + A343063(n, 3).
a(n) = A343064(n) + A343065(n) + A343066(n).
Showing 1-10 of 11 results. Next

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