OFFSET
1,1
COMMENTS
(a, b, c) is an ABC triple if gcd(a, b) = 1 and a + b = c. ABC triples with c > rad(a*b*c) are called "abc-hits". For primitive Pythagorean triples (u, v, w) it is u^2 + v^2 = w^2 and gcd(u^2, v^2) = 1. (u^2, v^2, w^2) are therefore ABC triples. They are then "abc-hits" if in addition w^2 > rad(u^2*v^2*w^2). If (u, v, w) is a non-primitive Pythagorean triple, (u^2, v^2, w^2) is not an ABC triple.
w of primitive Pythagorean triples (u, v, w) with A007947(u^2*v^2*w^2) < w^2.
LINKS
Felix Huber, Table of n, a(n) for n = 1..1420
Abderrahmane Nitaj, The ABC Conjecture Home Page
Wikipedia, abc conjecture
EXAMPLE
25 from the primitive Pythagorean triple (7, 24, 25) is in the sequence, because 7^2 + 24^2 = 25^2, gcd(7^2, 24^2) = 1 and 25^2 = 625 > rad(7^2*24^2*25^2) = 7*2*3*5 = 210.
MAPLE
A366428:=proc(N) # To get all terms <= N
local a, b, c, i, l, p, q;
l:=[];
for p from 2 to floor(sqrt(N-1)) do
for q to min(p-1, floor(sqrt(N-p^2))) do
if gcd(p, q)=1 and is(p-q, odd) then
c:=p^2+q^2;
b:=max(p^2-q^2, 2*p*q);
a:=min(p^2-q^2, 2*p*q);
if c^2>NumberTheory:-Radical(c*b*a) then
l:=[op(l), [c, b, a]];
fi
fi
od
od;
return seq((sort(l)[i])[1], i=1..nops(l));
end proc:
A366428(10985);
CROSSREFS
KEYWORD
nonn
AUTHOR
Felix Huber, Oct 13 2023
STATUS
approved
