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
KEYWORD
nonn,easy
AUTHOR
STATUS
approved