

A024365


Areas of right triangles with coprime integer sides.


17



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
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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 345 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 *)


PROG

(PARI) select( {is_A024365(n)=my(N=1+#n=divisors(2*n)); for(i=1, N\2, gcd(n[i], n[Ni])==1 && issquare(n[i]^2+n[Ni]^2) && return(n[i]))}, [1..10^4]) \\ is_A024365 returns the smaller leg if n is a term, else 0.  M. F. Hasler, Jun 06 2024


CROSSREFS



KEYWORD

nonn,easy


AUTHOR



EXTENSIONS



STATUS

approved



