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

1,1

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

Links

Ray Chandler, Table of n, a(n) for n = 1..10000

Ron Knott, Pythagorean Triples and Online Calculators

Example

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)

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 *)

Crossrefs

Cf. A020883.

Sequence in context: A061118 A097822 A069064 * A200521 A350374 A189982

Adjacent sequences: A024407 A024408 A024409 * A024411 A024412 A024413

Keyword

nonn

Author

David W. Wilson

Status

approved