

A024410


Long leg of more than one primitive Pythagorean triangle.


3



420, 572, 780, 840, 924, 1020, 1292, 1596, 1680, 1716, 1848, 1932, 2100, 2145, 2244, 2300, 2484, 2520, 2640, 2652, 2700, 2900, 2964, 3080, 3132, 3315, 3348, 3432, 3465, 3596, 3640, 3828, 3876, 3960, 4060, 4092, 4095, 4340, 4488, 4588, 4620, 4680, 4692
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

Also, middle side a of more than one primitive integersided triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c; hence, terms that appear more than once in A020883.  Bernard Schott, Oct 21 2021


LINKS



EXAMPLE

> For primitive Pythagorean triples:
a(1) = 420 because 420 is the smallest long leg that belongs to more than one primitive Pythagorean triples, we have 29^2 + 420^2 = 421^2 and 341^2 + 420^2 = 541^2.
> For primitive triples with 2/a = 1/b + 1/c:
a(1) = 420 because 420 is the smallest middle side a that belongs to more than one primitive integersided triangles (a, b, c) where side a is the harmonic mean of the 2 other sides b and c, i.e., 2/a = 1/b + 1/c with b < a < c, we have 2/420 = 1/310 + 1/651 and 2/420 = 1/406 + 1/435. (End)


MATHEMATICA

bb=1; s=e=""; For[b=1, b<=12^3, For[a=b1, a>2, c=(a^2+b^2)^0.5; If[c==Round[c]&&GCD[a, b]==1, If[b==bb, e=e<>ToString[b]<>", "; s=s<>ToString[a]<>", "<>ToString[b]<>", "<>ToString[Round[c]]<>"; "]; bb=b]; a ]; b++ ]; Print["B = ", e] (* Vladimir Joseph Stephan Orlovsky, Apr 29 2008 *)
Select[Tally[Union[Sort/@({Times@@#, (Last[#]^2First[#]^2)/2}&/@(Select[ Subsets[Range[1, 121, 2], {2}], GCD@@#==1&]))][[All, 2]]], #[[2]]>1&][[All, 1]] //Sort (* Harvey P. Dale, Mar 07 2020 *)


CROSSREFS



KEYWORD

nonn


AUTHOR



STATUS

approved



