|
|
A024365
|
|
Areas of right triangles with coprime integer sides.
|
|
16
|
|
|
6, 30, 60, 84, 180, 210, 330, 504, 546, 630, 840, 924, 990, 1224, 1320, 1386, 1560, 1710, 1716, 2310, 2340, 2574, 2730, 3036, 3570, 3900, 4080, 4290, 4620, 4914, 5016, 5610, 5814, 6090, 6630, 7140, 7440, 7854, 7956, 7980, 8970, 8976, 9690, 10374
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,1
|
|
COMMENTS
|
Consider primitive Pythagorean triangles (A^2 + B^2 = C^2, (A, B) = 1, A <= B); sequence gives areas A*B/2.
By Theorem 2 of Mohanty and Mohanty, all these numbers are primitive Pythagorean. - T. D. Noe, Sep 24 2013
This sequence also gives Fibonacci's congruous numbers (without multiplicity, in increasing order) divided by 4. See A258150. - Wolfdieter Lang, Jun 14 2015
|
|
LINKS
|
|
|
FORMULA
|
Positive integers of the form u*v*(u^2 - v^2) where 2uv and u^2 - v^2 are coprime or, alternatively, where u, v are coprime and one of them is even.
|
|
EXAMPLE
|
6 is in the sequence because it is the area of the 3-4-5 triangle.
a(7) = 210 corresponds to the two primitive Pythagorean triangles (21, 20, 29) and (35, 12, 37). See A024406. - Wolfdieter Lang, Jun 14 2015
|
|
MATHEMATICA
|
nn = 22; (* nn must be even *) t = Union[Flatten[Table[If[GCD[u, v] == 1 && Mod[u, 2] + Mod[v, 2] == 1, u v (u^2 - v^2), 0], {u, nn}, {v, u - 1}]]]; Select[Rest[t], # < nn (nn^2 - 1) &] (* T. D. Noe, Sep 19 2013 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
nonn,easy
|
|
AUTHOR
|
|
|
EXTENSIONS
|
|
|
STATUS
|
approved
|
|
|
|