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.

A366674 A366428 corresponding values for min(u, v) of Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit".

Original entry on oeis.org

7, 9, 16, 44, 17, 161, 175, 135, 297, 336, 52, 41, 116, 64, 49, 57, 720, 63, 276, 828, 96, 825, 237, 81, 1377, 1320, 128, 2016, 2080, 97, 3367, 99, 495, 721, 160, 1296, 164, 117, 5184, 125, 375, 127, 3375, 959, 824, 2793
Offset: 1

Views

Author

Felix Huber, Oct 16 2023

Keywords

Examples

			A366438(1) = 25, the corresponding primitive Pythagorean triple is (7, 24, 25). a(1) = min(7, 24) = 7.

Crossrefs

Cf. A366428.

A366675 A366428 corresponding values for max(u, v) of Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit".

Original entry on oeis.org

24, 40, 63, 117, 144, 240, 288, 352, 304, 527, 675, 840, 837, 1023, 1200, 1624, 1519, 1984, 2107, 2035, 2303, 2752, 3116, 3280, 3136, 3479, 4095, 3713, 3969, 4704, 3456, 4900, 4888, 5280, 6399, 6497, 6723, 6844, 5537, 7812, 7808, 8064, 7448, 9360, 10593, 10624
Offset: 1

Views

Author

Felix Huber, Oct 16 2023

Keywords

Examples

			A366438(1) = 25, the corresponding primitive Pythagorean triple is (7, 24, 25). a(1) = max(7, 24) = 24.

Crossrefs

Cf. A366428.

A386307 Ordered hypotenuses of Pythagorean triples that do not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), where u and v are positive integers.

Original entry on oeis.org

15, 25, 30, 35, 39, 50, 51, 55, 60, 65, 65, 70, 75, 75, 78, 85, 85, 87, 91, 95, 100, 102, 105, 110, 111, 115, 119, 120, 123, 125, 130, 130, 135, 140, 143, 145, 145, 150, 150, 155, 156, 159, 165, 169, 170, 170, 174, 175, 175, 182, 183, 185, 185, 187, 190, 195, 195
Offset: 1

Views

Author

Felix Huber, Aug 13 2025

Keywords

Comments

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2) is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.
A101930(n) gives the total number of Pythagorean triples <= 10^n. The percentage of triangles in this sequence increases continuously:
number of terms <= h total number of
h in this sequence hypotenuses <= h percentage
10 0 2 0.0 %
100 21 52 40.4 %
1000 514 881 58.3 %
10000 8629 12471 69.2 %
100000 122431 161436 75.8 %

Examples

			The Pythagorean triple (9, 12, 15) does not have the form (u^2 - v^2, 2*u*v, u^2 + v^2), because 15 is not a sum of two nonzero squares. Therefore 15 is a term.

Links

Crossrefs

Subsequence of A009000.
Cf. A101930, A366428, A380072, A386308, A386309, A386943.

Programs

  • Maple
    A386307:=proc(N) # To get all hypotenuses <= N
    local i,l,m,u,v,r,x,y,z;
    l:={};
    m:={};
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            x:=min(2*u*v,u^2-v^2);
            y:=max(2*u*v,u^2-v^2);
            z:=u^2+v^2;
            m:=m union {[z,y,x]};
            if gcd(u,v)=1 and is(u-v,odd) then
                l:=l union {seq([i*z,i*y,i*x],i=1..N/z)}
            fi
        od
    od;
    r:=l minus m;
    return seq(r[i,1],i=1..nops(r));
end proc;
A386307(1000);

Formula

a(n) = sqrt(A386308(n)^2 + A386309(n)^2).
{A009000(n)} = {a(n)} union {A020882(n)} union {A386943(n)}.

A386943 Ordered hypotenuses of nonprimitive Pythagorean triples of the form (u^2 - v^2, 2*u*v, u^2 + v^2), where u and v are positive integers.

Original entry on oeis.org

10, 20, 26, 34, 40, 45, 50, 52, 58, 68, 74, 80, 82, 90, 100, 104, 106, 116, 117, 122, 125, 130, 130, 136, 146, 148, 153, 160, 164, 170, 170, 178, 180, 194, 200, 202, 208, 212, 218, 225, 226, 232, 234, 244, 245, 250, 250, 260, 260, 261, 272, 274, 290, 290, 292, 296
Offset: 1

Views

Author

Felix Huber, Aug 24 2025

Keywords

Comments

In the form (u^2 - v^2, 2*u*v, u^2 + v^2), u^2 + v^2 is the hypotenuse, max(u^2 - v^2, 2*u*v) is the long leg and min(u^2 - v^2, 2*u*v) is the short leg.
A101930(n) gives the total number of Pythagorean triples <= 10^n.
number of terms <= h total number of
h in this sequence hypotenuses <= h percentage
10 1 2 50.0 %
100 15 52 28.8 %
1000 209 881 23.7 %
10000 2249 12471 18.0 %
100000 23086 161436 14.3 %

Examples

			The nonprimitive Pythagorean triple (6, 8, 10) is of the form (u^2 - v^2, 2*u*v, u^2 + v^2): From u = 3 and v = 1 follows u^2 - v^2 = 8 (long leg), 2*u*v = 6 (short leg), u^2 - v^2 = 10 (hypotenuse). Therefore, 10 is a term.

Links

Crossrefs

Subsequence of A009000.
Cf. A020882, A101930, A366428, A380072, A386307, A386944, A386945.

Programs

  • Maple
    A386943:=proc(N) # To get all hypotenuses <= N
    local i,l,u,v;
    l:=[];
    for u from 2 to floor(sqrt(N-1)) do
        for v to min(u-1,floor(sqrt(N-u^2))) do
            if gcd(u,v)>1 or is(u-v,even) then
                l:=[op(l),[u^2+v^2,max(2*u*v,u^2-v^2),min(2*u*v,u^2-v^2)]]
            fi
        od
    od;
    l:=sort(l);
    return seq(l[i,1],i=1..nops(l));
end proc;
A386943(296);

Formula

a(n) = sqrt(A386944(n)^2 + A386945(n)^2).
{A009000(n)} = {a(n)} union {A020882(n)} union {A386307(n)}.
Showing 1-4 of 4 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.