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A366428 Hypotenuse numbers w of Pythagorean triples (u, v, w) for which (u^2, v^2, w^2) is an "abc-hit". 3
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 (list; graph; refs; listen; history; text; internal format)
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.
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).
LINKS
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.
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).
Sequence in context: A255608 A309623 A242074 * A195564 A147287 A240758
KEYWORD
nonn
AUTHOR
Felix Huber, Oct 13 2023
STATUS
approved

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Last modified July 22 09:18 EDT 2024. Contains 374485 sequences. (Running on oeis4.)