

A198441


Square root of third term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).


12



7, 17, 23, 31, 41, 47, 49, 71, 73, 79, 89, 97, 103, 113, 119, 119, 127, 137, 151, 161, 161, 167, 191, 193, 199, 217, 217, 223, 233, 239, 241, 257, 263, 271, 281, 287, 287, 289, 311, 313, 329, 329, 337, 343, 353, 359, 367, 383, 391, 391, 401, 409, 431, 433
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OFFSET

1,1


COMMENTS

A198437(n) = a(n)^2; a(n) = A198390(A198409(n)).
This sequence gives the sum of the two legs (catheti) x + y of primitive Pythagorean triangles (x,y,z) with y even and gcd(x,y) = 1, ordered nondecreasingly (with multiple entries). See A058529(n), n>=2, for the sequence without multiple entries. For the proof, put in the Zumkeller link w = x + y, v = z and u = abs(x  y). This works because w^2  v^2 = v^2  u^2, hence u^2 = 2*v^2  w^2 = 2*z^2  (x+y)^2 = 2*(x^2 + y^2)  (x+y)^2 = x^2 + y^2  2*x*y = (xy)^2. The primitivity of the arithmetic progression triples follows from the one of the Pythagorean triples: gcd(u,w) = 1 follows from gcd(x,y) = 1, then gcd(u,v,w) = gcd(gcd(u,w),v) = 1. The converse can also be proved: given a primitive arithmetic progression triple (u,v,w), 1 <= u < v < w, gcd(u,v,w) = 1, the corresponding primitive Pythagorean triple with even y is ((wu)/2,(w+u)/2,v) or ((w+u)/2,(wu)/2,v), depending on whether (w+u)/2 is even or odd, respectively.  Wolfdieter Lang, May 22 2013
n appears A330174(n) times.  Ray Chandler, Feb 26 2020


LINKS

Ray Chandler, Table of n, a(n) for n = 1..10000
Reinhard Zumkeller, Table of initial values
Keith Conrad, Arithmetic progressions of three squares


EXAMPLE

Primitive Pythagorean triangle connection: a(1) = 7 because (u,v,w) = (1,5,7) corresponds to the primitive Pythagorean triangle (x = (wu)/2, y = (w+u)/2, z = v) = (3,4,5) with leg sum 3 + 4 = 7.  Wolfdieter Lang, May 23 2013


PROG

(Haskell)
a198441 n = a198441_list !! (n1)
a198441_list = map a198390 a198409_list


CROSSREFS

Cf. A225949 (triangle version of leg sums).
Cf. A198384, A198385, A198386.
Cf. A058529, A198390, A198409, A198437, A330174.
Sequence in context: A289363 A319040 A216838 * A058529 A253408 A120681
Adjacent sequences: A198438 A198439 A198440 * A198442 A198443 A198444


KEYWORD

nonn


AUTHOR

Reinhard Zumkeller, Oct 25 2011


STATUS

approved



