A366428 Hypotenuse numbers w of Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit".

25, 41, 65, 125, 145, 289, 337, 377, 425, 625, 677, 841, 845, 1025, 1201, 1625, 1681, 1985, 2125, 2197, 2305, 2873, 3125, 3281, 3425, 3721, 4097, 4225, 4481, 4705, 4825, 4901, 4913, 5329, 6401, 6625, 6725, 6845, 7585, 7813, 7817, 8065, 8177, 9409, 10625, 10985

Offset: 1

Author

Felix Huber, Oct 13 2023

Keywords

• nonn

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.

The corresponding values of min(u, v) and max(u, v) are in the sequences A366674 and A366675.

w of primitive Pythagorean triples (u, v, w) with A007947(u^2*v^2*w^2) < w^2.

Subsequence of intersection of A020882 and sqrt(A130510).

Examples

			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.
		

Links

• Felix Huber, Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit"
• Abderrahmane Nitaj, The ABC Conjecture Home Page
• Wikipedia, abc conjecture

Crossrefs

Cf. A366674, A366675 (corresponding values of min(u, v) and max(u, v)).

Cf. A020882 (hypotenuses of primitive Pythagorean triangles), A130510 ("abc-hits"), A007947 (squarefree kernel).