A198439 Square root of first term of a triple of squares in arithmetic progression that is not a multiple of another triple in (A198384, A198385, A198386).

1, 7, 7, 17, 1, 23, 31, 49, 17, 47, 23, 71, 7, 41, 41, 79, 97, 7, 31, 73, 127, 119, 89, 17, 161, 47, 113, 167, 119, 1, 199, 49, 73, 103, 161, 223, 241, 23, 31, 103, 89, 191, 287, 151, 217, 287, 137, 233, 71, 337, 79, 137, 17, 281, 359, 391, 49, 113, 119, 217

Offset: 1

Author

Reinhard Zumkeller, Oct 25 2011

Keywords

• nonn

Comments

This sequence gives the values |x-y| of primitive Pythagorean triangles (x,y,z) with even y ordered according to the nondecreasing values of the leg sums x+y (called w in the Zumkeller link, and given in A198441). For the equivalence to primitive Pythagorean triples with even y see a comment in A198441. - Wolfdieter Lang, May 22 2013

Examples

			From _Wolfdieter Lang_, May 22 2013: (Start)
Primitive Pythagorean triple (x,y,z), y even, connection:
a(2) = 7 because the triple with second smallest leg sum x+y = 17 = A198441(2) is (5,12,13), and |x - y| = y - x = 12 - 5 = 7.
a(3) = 7 because x + y = A198441(3) = 23, (x,y,z) = (15,8,17) (the primitive triple with third smallest leg sum), and |x-y| = x - y = 15 - 8 = 7. (End)
		

Links

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

Programs

Haskell

a198439 n = a198439_list !! (n-1)
a198439_list = map a198388 a198409_list

Mathematica

wmax = 1000;
triples[w_] := Reap[Module[{u, v}, For[u = 1, u < w, u++, If[IntegerQ[v = Sqrt[(u^2 + w^2)/2]], Sow[{u, v, w}]]]]][[2]];
tt = Flatten[DeleteCases[triples /@ Range[wmax], {}], 2];
DeleteCases[tt, t_List /; GCD@@t>1 && MemberQ[tt, t/GCD@@t]][[All, 1]] (* Jean-François Alcover, Oct 22 2021 *)

Formula

A198435(n) = a(n)^2; a(n) = A198388(A198409(n)).