A024410 Long leg of more than one primitive Pythagorean triangle.

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

Offset: 1

Author

David W. Wilson

Keywords

• nonn

Comments

Also, middle side a of more than one primitive integer-sided 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

Examples

			From _Bernard Schott_, Oct 21 2021: (Start)
-> 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 integer-sided 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)
		

Links

• Ray Chandler, Table of n, a(n) for n = 1..10000
• Ron Knott, Pythagorean Triples and Online Calculators

Crossrefs

Cf. A020883.

Programs

Mathematica

bb=1;s=e="";For[b=1,b<=12^3,For[a=b-1,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[#]^2-First[#]^2)/2}&/@(Select[ Subsets[Range[1,121,2],{2}],GCD@@#==1&]))][[All,2]]],#[[2]]>1&][[All,1]] //Sort (* Harvey P. Dale, Mar 07 2020 *)