A379830 a(n) is the number of Pythagorean triples (u, v, w) for which w - u = n where u < v < w.

0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 1, 0, 1, 0, 1, 0, 2, 0, 4, 0, 1, 0, 1, 0, 2, 3, 1, 2, 1, 0, 1, 0, 5, 0, 1, 0, 4, 0, 1, 0, 2, 0, 1, 0, 1, 2, 1, 0, 2, 4, 7, 0, 1, 0, 4, 0, 2, 0, 1, 0, 1, 0, 1, 2, 5, 0, 1, 0, 1, 0, 1, 0, 8, 0, 1, 3, 1, 0, 1, 0, 2, 6, 1, 0, 1, 0, 1, 0

Offset

0,9

Comments

The difference between the hypotenuse and the short leg of a primitive Pythagorean triple (p^2 - q^2, 2*p*q, p^2 + q^2) (where p > q are coprimes and not both odd) is d = max(2*q^2, (p - q)^2). For every of these primitive Pythagorean triples whose d divides n, there is a Pythagorean triple with w - u = n. Therefore d <= n and it follows that 1 <= q <= sqrt(n/2) and q + 1 <= p <= q + sqrt(n), which means that there is a finite number of Pythagorean triples with w - u = n.

Links

Felix Huber, Table of n, a(n) for n = 0..10000

Felix Huber, Pythagorean triples for which w - u = n

Eric Weisstein's World of Mathematics, Pythagorean Triple

Example

The a(18) = 4 Pythagorean triples are (27, 36, 45), (16, 30, 34), (40, 42, 58), (7, 24, 25) because 45 - 27 = 34 - 16 = 58 - 40 = 25 - 7 = 18.
See also linked Maple program "Pythagorean triples for which w - u = n".

Maple

A379830:=proc(n)
    local a, p, q;
    a:=0;
    for q to isqrt(floor(n/2)) do
        for p from q+1 to q+isqrt(n) do
            if igcd(p, q)=1 and (is(p, even) or is(q, even)) and n mod max((p-q)^2, 2*q^2)=0 then
                a:=a+1
            fi
        od
    od;
    return a
end proc;
seq(A379830(n), n=0..87);

Crossrefs

Cf. A024361, A024362, A024363, A046079, A046080, A046081, A096033, A224921.

Sequence in context: A368948 A368579 A287331 * A179769 A340594 A361685

Adjacent sequences: A379827 A379828 A379829 * A379831 A379832 A379833

Keyword

nonn

Author

Felix Huber, Jan 07 2025

Status

approved