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A009112
Areas of Pythagorean triangles: numbers which can be the area of a right triangle with integer sides.
28
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
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
A073120 is a subsequence.
See A256418 for the numbers 4*a(n).
Sequence in context: A185210 A046131 A009111 * A057101 A057228 A334788
KEYWORD
nonn,easy
STATUS
approved