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

%I A009112 #86 Jan 05 2025 19:51:34
%S A009112 6,24,30,54,60,84,96,120,150,180,210,216,240,270,294,330,336,384,480,
%T A009112 486,504,540,546,600,630,720,726,750,756,840,864,924,960,990,1014,
%U A009112 1080,1176,1224,1320,1344,1350,1386,1470,1500,1536,1560,1620,1710,1716,1734,1890
%N A009112 Areas of Pythagorean triangles: numbers which can be the area of a right triangle with integer sides.
%C A009112 Number of terms < 10^k for increasing values of k: 1, 7, 34, 150, 636, 2536, 9757, 35987, 125350, 407538, ..., .
%C A009112 All terms are divisible by 6.
%C A009112 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
%H A009112 Robert Israel, <a href="/A009112/b009112.txt">Table of n, a(n) for n = 1..10000</a>
%H A009112 Ron Knott, <a href="http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Pythag/pythag.html#periarea">Pythagorean Triangles</a>
%H A009112 B. Miller, <a href="http://www.jstor.org/stable/27962235">Nasty Numbers</a>, The Mathematics Teacher 73 (1980), page 649.
%H A009112 Supriya Mohanty and S. P. Mohanty, <a href="https://web.archive.org/web/2024*/https://www.fq.math.ca/Scanned/28-1/mohanty.pdf">Pythagorean Numbers</a>, Fibonacci Quarterly 28 (1990), 31-42.
%e A009112 30 belongs to the sequence as the area of the triangle (5,12,13) is 30.
%e A009112 6 is in the sequence because it is the area of the 3-4-5 triangle.
%p A009112 N:= 10^4: # to get all entries <= N
%p A009112 A:= {}:
%p A009112 for t from 1 to floor(sqrt(2*N)) do
%p A009112    F:= select(f -> f[2]::odd,ifactors(2*t)[2]);
%p A009112    d:= mul(f[1],f=F);
%p A009112    for e from ceil(sqrt(t/d)) do
%p A009112      s:= d*e^2;
%p A009112      r:= sqrt(2*t*s);
%p A009112      a:= (r+s)*(r+t)/2;
%p A009112      if a > N then break fi;
%p A009112      A:= A union {a};
%p A009112    od
%p A009112 od:
%p A009112 A;
%p A009112 # if using Maple 11 or earlier, uncomment the next line
%p A009112 # sort(convert(A,list)); # _Robert Israel_, Apr 06 2015
%t A009112 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 *)
%t A009112 g@A_ := With[{div = Divisors@(2*A)}, AnyTrue[Sqrt@(Plus@@({#, 2*A/#}^2))& /@Take[div, Floor[(Length@div)/2]],IntegerQ]];
%t A009112 Select[Range@5000, g@# &] (* _Hans Rudolf Widmer_, Sep 25 2023 *)
%o A009112 (PARI) 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
%o A009112 (Sage) 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
%Y A009112 Union of A009111, A009127, A024365, A177021.
%Y A009112 A073120 is a subsequence.
%Y A009112 See A256418 for the numbers 4*a(n).
%K A009112 nonn,easy
%O A009112 1,1
%A A009112 _David W. Wilson_