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

1,1

Comments

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

Links

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.

Example

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.

Maple

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

Mathematica

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 *)

Prog

(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
(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

Crossrefs

Union of A009111, A009127, A024365, A177021.

A073120 is a subsequence.

See A256418 for the numbers 4*a(n).

Sequence in context: A185210 A046131 A009111 * A057101 A057228 A334788

Adjacent sequences: A009109 A009110 A009111 * A009113 A009114 A009115

Keyword

nonn,easy

Author

David W. Wilson

Status

approved