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.

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

1,1

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 %

Links

Felix Huber, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Pythagorean Triple

Formula

a(n) = sqrt(A386944(n)^2 + A386945(n)^2).

{A009000(n)} = {a(n)} union {A020882(n)} union {A386307(n)}.

Example

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.

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);

Crossrefs

Subsequence of A009000.

Cf. A020882, A101930, A366428, A380072, A386307, A386944, A386945.

Sequence in context: A072592 A229305 A229304 * A101869 A015872 A019491

Adjacent sequences: A386940 A386941 A386942 * A386944 A386945 A386946

Keyword

nonn,easy

Author

Felix Huber, Aug 24 2025

Status

approved