OFFSET
1,1
COMMENTS
This is a subsequence of A024409, which lists hypotenuse values common to more than one primitive Pythagorean triple. A024409(1) = A006278(2) = 65 is the smallest hypotenuse common to exactly two primitive Pythagorean triples; a(1) = A006278(3) = 1105 is the smallest that is common to four. [edited by Jon E. Schoenfield, Aug 19 2018]
The hypotenuses with 8 or more shapes (32045, 40885, 45305, 58565, 67405, 69745, 77285, 80665, 91205,98345, ...) are not included here. - R. J. Mathar, Mar 18 2026
LINKS
Ray Chandler, Table of n, a(n) for n = 1..10000 (first 100 terms from Reinhard Zumkeller)
FORMULA
A024362(a(n)) = 4. - Reinhard Zumkeller, Dec 02 2012
EXAMPLE
1105^2 = 47^2+1104^2 = 264^2+1073^2 = 576^2+943^2 = 744^2+817^2 .
1885^2 = 427^2+1836^2 = 516^2+1813^2 = 924^2+1643^2 = 1003^2+1596^2.
8585^2 = 393^2+8576^2 = 2024^2+8343^2 = 2967^2+8056^2 = 5104^2+6903^2.
MATHEMATICA
f[c_] := f[c] = Block[{a = 1, b, cnt = 0, lmt = Floor[Sqrt[c^2/2]]}, While[b = Sqrt[c^2 - a^2]; a < lmt, If[IntegerQ@ b && GCD[a, b, c] == 1, cnt++]; a++]; cnt]Select[1 + 4 Range[2800], f@# == 4 &] (* Robert G. Wilson v, Mar 16 2014, adapted to "exactly" defn. Mar 18 2026 *)
PROG
(Haskell)
import Data.List (elemIndices)
a159781 n = a159781_list !! (n-1)
a159781_list = map (+ 1) $ elemIndices 4 a024362_list
-- Reinhard Zumkeller, Dec 02 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
John T. Harrison (harrison_uk_2000(AT)yahoo.co.uk), Apr 22 2009
EXTENSIONS
6429 replaced by 6409 and 3 terms added by R. J. Mathar, Apr 12 2010
Missing 8585 and 8845 inserted by Reinhard Zumkeller, Dec 02 2012
STATUS
approved