A177021 -id:A177021 - cp's OEIS frontend

A009112 Areas of Pythagorean triangles: numbers which can be the area of a right triangle with integer sides.

6, 24, 30, 54, 60, 84, 96, 120, 150, 180, 210, 216, 240, 270, 294, 330, 336, 384, 480, 486, 504, 540, 546, 600, 630, 720, 726, 750, 756, 840, 864, 924, 960, 990, 1014, 1080, 1176, 1224, 1320, 1344, 1350, 1386, 1470, 1500, 1536, 1560, 1620, 1710, 1716, 1734, 1890

Offset: 1

David W. Wilson

Number of terms < 10^k for increasing values of k: 1, 7, 34, 150, 636, 2536, 9757, 35987, 125350, 407538, ..., .
All terms are divisible by 6.
Also positive integers m with four (or more) different divisors (p, q, r, s) such that m = p*q = r*s and s = p+q+r. - Jose Aranda, Jun 28 2023

			30 belongs to the sequence as the area of the triangle (5,12,13) is 30.
6 is in the sequence because it is the area of the 3-4-5 triangle.

Robert Israel, Table of n, a(n) for n = 1..10000
Ron Knott, Pythagorean Triangles
B. Miller, Nasty Numbers, The Mathematics Teacher 73 (1980), page 649.
Supriya Mohanty and S. P. Mohanty, Pythagorean Numbers, Fibonacci Quarterly 28 (1990), 31-42.

Union of A009111, A009127, A024365, A177021.
A073120 is a subsequence.
See A256418 for the numbers 4*a(n).

N:= 10^4: # to get all entries <= N
A:= {}:
for t from 1 to floor(sqrt(2*N)) do
   F:= select(f -> f[2]::odd,ifactors(2*t)[2]);
   d:= mul(f[1],f=F);
   for e from ceil(sqrt(t/d)) do
     s:= d*e^2;
     r:= sqrt(2*t*s);
     a:= (r+s)*(r+t)/2;
     if a > N then break fi;
     A:= A union {a};
   od
od:
A;
# if using Maple 11 or earlier, uncomment the next line
# sort(convert(A,list)); # Robert Israel, Apr 06 2015

lst = {}; Do[ If[ IntegerQ[c = Sqrt[a^2 + b^2]], AppendTo[lst, a*b/2]; lst = Union@ lst], {a, 4, 180}, {b, a - 1, Floor[ Sqrt[a]], -1}]; Take[lst, 51] (* Vladimir Joseph Stephan Orlovsky, Nov 23 2010 *)
g@A_ := With[{div = Divisors@(2*A)}, AnyTrue[Sqrt@(Plus@@({#, 2*A/#}^2))& /@Take[div, Floor[(Length@div)/2]],IntegerQ]];
Select[Range@5000, g@# &] (* Hans Rudolf Widmer, Sep 25 2023 *)

is_A009112(n)={ my(N=1+#n=divisors(2*n)); for( i=1, N\2, issquare(n[i]^2+n[N-i]^2) & return(1)) } \\ M. F. Hasler, Dec 09 2010

is_A009112 = lambda n: any(is_square(a**2+(2*n/a)**2) for a in divisors(2*n)) # D. S. McNeil, Dec 09 2010

A177063 Number of Pythagorean triangles with area n.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1

Offset: 1

M. F. Hasler, Dec 09 2010

nonn

The first term > 1 is a(210) = 2, cf. A009127, A055193 and A024407. Up to there the sequence coincides with the characteristic function of A009112. The triangles are not necessarily primitive.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries related to Pythagorean triples.

Cf. A009111, A009127, A177021.

a(n)=my(N=1+#n=divisors(2*n));sum(i=1,N\2,issquare(n[i]^2+n[N-i]^2));

Secondary offset added by Antti Karttunen, Nov 24 2017

A291420 Numbers n such that there exist exactly four distinct Pythagorean triangles, at least one of them primitive, with area n.

Original entry on oeis.org

341880, 8168160, 14636160, 17957940, 52492440, 116396280, 1071572040, 1187525640, 1728483120, 5988702720, 6609482880, 22539095040, 29239970760, 136496680320, 258670630680, 398648544840, 494892478080, 592003418160, 1329673884000, 1343798407560, 2190884461920

Offset: 1

Sture Sjöstedt, Aug 23 2017

nonn

Numbers n such that there exist positive integers x, y with x > y and n = x*y*(x-y)*(x+y).

Many of them consist of a Pythagorean triangle plus a triple which is a solution to Carroll's problem: Find three Pythagorean triangles with the same area.

			p^2 - p*q + q^2 = r^2;
p = 208, q = 418, r = 362, q - p = 210;
n = p*r*q*(q-p) = 208*418*362*210 = 6609482880.
x = 640, y = 627 gives the same area:
n = x*y*(x-y)*(x+y) = 640*627*13*1267 = 6609482880.

Cf. A009127, A024407, A055193, A088513, A088977, A089025, A177021, A291591.

a(12)-a(21) from Giovanni Resta, Aug 28 2017

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A009112 Areas of Pythagorean triangles: numbers which can be the area of a right triangle with integer sides.

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A177063 Number of Pythagorean triangles with area n.

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A291420 Numbers n such that there exist exactly four distinct Pythagorean triangles, at least one of them primitive, with area n.

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