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.

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A196098 Positive integers a for which there is a (5/4)-Pythagorean triple (a,b,c) satisfying a<=b.

Original entry on oeis.org

3, 4, 6, 8, 9, 10, 11, 12, 12, 14, 15, 15, 16, 17, 18, 19, 20, 20, 20, 21, 22, 22, 23, 24, 24, 25, 26, 27, 28, 28, 28, 29, 30, 30, 30, 31, 32, 32, 33, 33, 33, 34, 35, 35, 36, 36, 36, 37, 38, 39, 40, 40, 40, 42, 42, 42, 44, 44, 44, 45, 45, 46, 47, 48, 48, 50, 51, 51
Offset: 1

Views

Author

Clark Kimberling, Sep 28 2011

Keywords

Comments

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

Crossrefs

Cf. A195770, A196097, A196099, A196100, A196101.

Programs

  • Mathematica
    z8 = 600; z9 = 150; z7 = 100;
k = 5/3; c[a_, b_] := Sqrt[a^2 + b^2 + k*a*b];
d[a_, b_] := If[IntegerQ[c[a, b]], {a, b, c[a, b]}, 0]
t[a_] := Table[d[a, b], {b, a, z8}]
u[n_] := Delete[t[n], Position[t[n], 0]]
Table[u[n], {n, 1, 15}]
t = Table[u[n], {n, 1, z8}];
Flatten[Position[t, {}]]
u = Flatten[Delete[t, Position[t, {}]]];
x[n_] := u[[3 n - 2]];
Table[x[n], {n, 1, z7}]  (* A196088 *)
y[n_] := u[[3 n - 1]];
Table[y[n], {n, 1, z7}]  (* A196089 *)
z[n_] := u[[3 n]];
Table[z[n], {n, 1, z7}]  (* A196090 *)
x1[n_] := If[GCD[x[n], y[n], z[n]] == 1, x[n], 0]
y1[n_] := If[GCD[x[n], y[n], z[n]] == 1, y[n], 0]
z1[n_] := If[GCD[x[n], y[n], z[n]] == 1, z[n], 0]
f = Table[x1[n], {n, 1, z9}];
x2 = Delete[f, Position[f, 0]]  (* A196091 *)
g = Table[y1[n], {n, 1, z9}];
y2 = Delete[g, Position[g, 0]]  (* A196092 *)
h = Table[z1[n], {n, 1, z9}];
z2 = Delete[h, Position[h, 0]]  (* A196093 *)

A196110 Positive integers a for which there is a primitive (7/4)-Pythagorean triple (a,b,c) satisfying a<=b.

Original entry on oeis.org

76, 47, 38, 34, 32, 31, 271, 113, 83, 586, 317, 184, 158, 61, 109, 383, 257, 827, 578, 92, 454, 331, 139, 212, 527, 844, 698, 601, 241, 409, 362, 893, 316, 197, 272, 391, 631, 233, 398, 541, 901, 782, 664, 214, 548, 437, 886, 527, 722, 1009, 323, 961
Offset: 1

Views

Author

Clark Kimberling, Sep 28 2011

Keywords

Comments

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

Crossrefs

Cf. A195770, A196105, A196108, A196109.

Programs

A196112 Positive integers a for which there is a 3-Pythagorean triple (a,b,c) satisfying a<=b.

Original entry on oeis.org

3, 5, 6, 7, 8, 9, 9, 10, 11, 11, 12, 13, 13, 14, 15, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 20, 21, 21, 21, 21, 21, 22, 22, 23, 23, 24, 24, 24, 25, 25, 26, 26, 27, 27, 27, 27, 28, 29, 30, 30, 30, 31, 32, 32, 32, 33, 33, 33, 33, 33, 34, 34, 35, 35, 36, 36, 37, 38, 38
Offset: 1

Views

Author

Clark Kimberling, Sep 28 2011

Keywords

Comments

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

Crossrefs

Cf. A195770, A196113, A196114, A196115.

Programs

  • Mathematica
    z8 = 900; z9 = 250; z7 = 200;
k = 7/4; c[a_, b_] := Sqrt[a^2 + b^2 + k*a*b];
d[a_, b_] := If[IntegerQ[c[a, b]], {a, b, c[a, b]}, 0]
t[a_] := Table[d[a, b], {b, a, z8}]
u[n_] := Delete[t[n], Position[t[n], 0]]
Table[u[n], {n, 1, 15}]
t = Table[u[n], {n, 1, z8}];
Flatten[Position[t, {}]]
u = Flatten[Delete[t, Position[t, {}]]];
x[n_] := u[[3 n - 2]];
Table[x[n], {n, 1, z7}]  (* A196105 *)
y[n_] := u[[3 n - 1]];
Table[y[n], {n, 1, z7}]  (* A196106 *)
z[n_] := u[[3 n]];
Table[z[n], {n, 1, z7}]  (* A196107 *)
x1[n_] := If[GCD[x[n], y[n], z[n]] == 1, x[n], 0]
y1[n_] := If[GCD[x[n], y[n], z[n]] == 1, y[n], 0]
z1[n_] := If[GCD[x[n], y[n], z[n]] == 1, z[n], 0]
f = Table[x1[n], {n, 1, z9}];
x2 = Delete[f, Position[f, 0]]  (* A196108 *)
g = Table[y1[n], {n, 1, z9}];
y2 = Delete[g, Position[g, 0]]  (* A196109 *)
h = Table[z1[n], {n, 1, z9}];
z2 = Delete[h, Position[h, 0]]  (* A196110 *)

A196174 Positive integers c for which there is a primitive 7-Pythagorean triple (a,b,c) satisfying a<=b.

Original entry on oeis.org

3, 11, 19, 101, 29, 93, 281, 33, 59, 109, 183, 551, 31, 41, 57, 179, 181, 911, 87, 149, 271, 453, 123, 209, 379, 633, 71, 139, 237, 719, 61, 213, 359, 649, 79, 267, 449, 811, 89, 121, 393, 659, 319, 401, 779, 199, 627, 229, 717, 177, 311, 571, 957, 131
Offset: 1

Views

Author

Clark Kimberling, Sep 29 2011

Keywords

Comments

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

Crossrefs

Cf. A195770, A196169, A196177, A196173.

Programs

A196186 Positive integers a for which there is a primitive 9-Pythagorean triple (a,b,c) satisfying a<=b.

Original entry on oeis.org

1, 3, 3, 3, 3, 5, 5, 5, 7, 8, 8, 8, 9, 9, 11, 13, 13, 15, 15, 15, 15, 15, 16, 16, 16, 17, 17, 19, 19, 21, 21, 23, 23, 24, 24, 24, 24, 25, 27, 29, 31, 32, 32, 33, 35, 37, 39, 39, 39, 39, 40, 40, 40, 41, 43, 45, 45, 45, 47, 48, 48, 51, 51, 51, 53, 55, 56
Offset: 1

Views

Author

Clark Kimberling, Sep 29 2011

Keywords

Comments

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

Examples

			Primitive 9-Pythagorean triples a,b,c;
c^2=a^2+b^2+9*a*b:
1,15,19
3,5,13
3,8,17
3,13,23
3,160,173
5,24,41
5,48,67
5,459,481
7,57,83
8,15,37

Crossrefs

Cf. A195770, A196183, A196187, A196188.

Programs

A196249 Positive integers a for which there is a primitive (-3/4)-Pythagorean triple (a,b,c) satisfying a<=b.

Original entry on oeis.org

3, 3, 4, 6, 7, 9, 10, 11, 12, 14, 17, 20, 21, 22, 23, 25, 26, 28, 29, 30, 31, 32, 32, 32, 34, 35, 36, 37, 38, 39, 41, 42, 49, 52, 53, 55, 57, 58, 60, 61, 63, 64, 64, 68, 69, 70, 74, 76, 76, 82, 84, 84, 84, 87, 91, 91, 92, 92, 93, 95, 96, 96, 96, 96, 96, 98, 99, 101
Offset: 1

Views

Author

Clark Kimberling, Sep 30 2011

Keywords

Comments

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

Examples

			Primitive (-3/4)-Pythagorean triples a,b,c;
c^2=a^2+b^2+k*a*b, where k=-3/4:
3,4,4
3,32,31
4,15,14
6,14,13
7,36,34
9,28,26
10,18,17
11,420,416
14,342,337

Crossrefs

Cf. A195770, A196245, A196249, A196250.

Programs

A196255 Positive integers a for which there is a primitive (3/4)-Pythagorean triple (a,b,c) satisfying a<=b.

Original entry on oeis.org

2, 5, 7, 9, 12, 12, 13, 15, 17, 18, 20, 21, 23, 26, 27, 28, 28, 31, 32, 32, 33, 33, 36, 38, 39, 42, 42, 43, 44, 45, 47, 51, 52, 54, 57, 59, 60, 62, 64, 64, 65, 66, 67, 68, 70, 77, 78, 83, 84, 87, 91, 94, 96, 96, 96, 96, 100, 102, 105, 107, 111, 116, 117, 117, 123
Offset: 1

Views

Author

Clark Kimberling, Sep 30 2011

Keywords

Comments

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

Examples

			Primitive (3/4)-Pythagorean triples a,b,c;
c^2=a^2+b^2+k*a*b, where k=3/4:
2,6,7
5,84,86
7,12,16
9,59,56
12,19,26
12,119,124
13,576,581
15,64,71
17,192,199
18,550,557

Crossrefs

Cf. A195770, A196252, A196256, A196257.

Programs

A196262 Positive integers a in primitive (1/4)-Pythagorean triples (a,b,c) satisfying a<=b, in order of increasing a and then increasing b.

Original entry on oeis.org

2, 4, 5, 6, 7, 7, 9, 10, 11, 12, 13, 15, 15, 17, 19, 20, 20, 22, 22, 23, 23, 25, 25, 26, 28, 29, 31, 31, 32, 32, 32, 33, 34, 36, 37, 38, 38, 39, 39, 41, 41, 43, 44, 44, 46, 47, 47, 50, 52, 52, 52, 53, 54, 55, 55, 55, 55, 57, 58, 58, 59, 60, 61, 62, 63, 64, 64, 64, 64, 65, 65, 65, 67, 68, 68, 68
Offset: 1

Views

Author

Clark Kimberling, Sep 30 2011

Keywords

Comments

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

Examples

			Primitive (1/4)-Pythagorean triples a,b,c where c^2=a^2+b^2+(1/4)*a*b:
  2,  2,  3
  4, 15, 16
  5, 32, 33
  6, 70, 71
  7, 20, 22
  7,192,193
  9, 44, 46
 10, 26, 29
 11, 20, 24
 12, 17, 22

Links

Crossrefs

Cf. A195770, A196259, A196263, A196264.

Programs

  • Maple
    F:= proc(a)
  sort(select(t -> subs(t, b) >= a and subs(t, c) > 0 and igcd(a, subs(t,b),subs(t,c)) = 1, [isolve](4*a^2 + 4*b^2 + a*b = 4*c^2)), (s, t) -> subs(s, b) <= subs(t, b))
end proc:
seq(a$nops(F(a)),a=1..100);# Robert Israel, Dec 20 2024
  • Mathematica
    (See A196259.)

Extensions

Corrected by Robert Israel, Dec 20 2024

A196263 Positive integers b in primitive (1/4)-Pythagorean triples (a,b,c) satisfying a<=b, in order of increasing a and then increasing b.

Original entry on oeis.org

2, 15, 32, 70, 20, 192, 44, 26, 20, 17, 220, 32, 884, 160, 64, 39, 391, 102, 950, 228, 2080, 32, 348, 186, 253, 1100, 416, 3780, 55, 77, 247, 608, 754, 1271, 1792, 310, 2838, 64, 5984, 96, 940, 340, 265, 629, 190, 960, 8692, 1634, 115, 287, 2655, 3680, 5734, 84, 468, 1316, 11904, 1820, 90, 938
Offset: 1

Views

Author

Clark Kimberling, Sep 30 2011

Keywords

Comments

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

Links

Crossrefs

Cf. A195770, A196259, A196262, A196264, A196260.

Programs

  • Maple
    F:= proc(a)
  sort(select(t -> subs(t, b) >= a and subs(t, c) > 0 and igcd(a, subs(t,b),subs(t,c)) = 1, [isolve](4*a^2 + 4*b^2 + a*b = 4*c^2)), (s, t) -> subs(s, b) <= subs(t, b))
end proc:
seq(op(map(t -> subs(t, b), F(a))), a=1..100); # Robert Israel, Dec 20 2024
  • Mathematica
    (See A196259.)

Extensions

Corrected by Robert Israel, Dec 25 2024

A196264 Positive integers c in primitive (1/4)-Pythagorean triples (a,b,c) satisfying a<=b, in order of increasing a and then increasing b.

Original entry on oeis.org

3, 16, 33, 71, 22, 193, 46, 29, 24, 22, 222, 37, 886, 163, 69, 46, 394, 107, 953, 232, 2083, 43, 352, 191, 258, 1104, 421, 3784, 67, 87, 253, 613, 759, 1276, 1797, 317, 2843, 79, 5989, 109, 946, 348, 274, 636, 201, 967, 8698, 1641, 132, 298, 2662, 3687, 5741, 106, 478, 1324, 11911, 1828, 113, 947
Offset: 1

Views

Author

Clark Kimberling, Sep 30 2011

Keywords

Comments

See A195770 for definitions of k-Pythagorean triple, primitive k-Pythagorean triple, and lists of related sequences.

Links

Crossrefs

Cf. A195770, A196259, A196262, A196263, A196261.

Programs

  • Maple
    F:= proc(a)
  sort(select(t -> subs(t, b) >= a and subs(t, c) > 0 and igcd(a, subs(t, b), subs(t, c)) = 1, [isolve](4*a^2 + 4*b^2 + a*b = 4*c^2)), (s, t) -> subs(s, b) <= subs(t, b))
end proc:
seq(op(map(t -> subs(t, c), F(a))), a=1..100); # Robert Israel, Dec 25 2024
  • Mathematica
    (See A196259.)

Extensions

Corrected by Robert Israel, Dec 25 2024
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