A393290 Primitive congruent numbers that are the squarefree part of the area of primitive Pythagorean triangles (PPTs) ordered within a Berggren ternary tree of PPTs.

6, 30, 210, 15, 21, 330, 70, 390, 1785, 154, 231, 65, 210, 5, 1155, 2730, 2730, 1254, 7854, 1995, 161, 286, 1254, 15834, 2990, 12369, 60639, 5610, 7, 1595, 4290, 3570, 154, 3135, 1610, 8866, 4641, 5610, 10374, 14, 330, 1330, 221, 4466, 158730, 6555, 3570, 98670, 741, 4030, 436254, 45066

Offset

1,1

Comments

A Berggren ternary tree is an infinite ordering of all PPTs with (3,4,5) as root node. The area of each PPT within this ternary tree has a primitive congruent number as its squarefree part. The Berggren tree ordering is as follows:

(3)

(4)

(5)

__________/ | \__________

/ | \

( 5) (21) (15)

(12) (20) ( 8)

(13) (29) (17)

/ | \ / | \ / | \

( 7) (55) (45) (39) (119) (77) (33) (65) (35)

(24) (48) (28) (80) (120) (36) (56) (72) (12)

(25) (73) (53) (89) (169) (85) (65) (97) (37)

/|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\ /|\

* * ** * ** * * * * ** * ** * * * * ** * ** * *

Links

Table of n, a(n) for n=1..52.

B. Berggren, Pytagoreiska Trianglar, Tidskrift för elementär matematik, fysik och kemi 17 (1934), 129-139 (English translation).

Frank M Jackson, Mathematica program

Wikipedia, Tree of primitive Pythagorean triples.

Example

The sequence of primitive congruent numbers that aligns with the Berggren ternary tree is as follows:
                          6
            __________/   |   \__________
           /              |              \
         30              210              15
     /   |   \        /   |   \        /   |   \
    21  330  70     390 1785  154    231  65   210
   /|\  /|\  /|\    /|\  /|\  /|\    /|\  /|\  /|\
  * * ** * ** * *  * * ** * ** * *  * * ** * ** * *
a(12) = 65. It is the squarefree part of area 2340 of PPT (65,72,97) because 2340 = 65*6^2 and 65 is a primitive congruent number.

Mathematica

(* See link above *)

Crossrefs

Cf. A006991. The squarefree part of A322170.

Sequence in context: A325950 A275953 A391604 * A322170 A362375 A369959

Adjacent sequences: A393287 A393288 A393289 * A393291 A393292 A393293

Keyword

nonn,tabf

Author

Frank M Jackson, Feb 09 2026

Status

approved