A363052 Integers m for which there exist positive integers j, k such that j*k*(j+k) = m^2.

4, 18, 24, 32, 36, 50, 60, 108, 140, 144, 150, 192, 252, 256, 288, 300, 360, 392, 400, 480, 486, 500, 540, 588, 648, 780, 816, 864, 882, 900, 972, 1008, 1014, 1050, 1120, 1152, 1156, 1176, 1200, 1350, 1372, 1452, 1536, 1620, 1764, 1800, 1848, 2016, 2040, 2048, 2178

Offset: 1

Author

Zhining Yang, May 15 2023

Keywords

• nonn
• easy

Comments

All terms are even.

Examples

			24 is a term: j*k*(j+k) = 24^2 for j=2, k=16.
		

Crossrefs

Cf. A088915.

Programs

Mathematica

Select[2*Range@500,
 Length@Select[Table[(Sqrt[b^2 + 4 #^2/b] - b)/2, {b, #}], IntegerQ] >
    0 &]
Select[Union@
  Flatten@Table[Sqrt[a*b (a + b)], {a, 1, 80}, {b, a, 500}],
 IntegerQ[#] && # < 1000 &]

Python

from itertools import count, islice
from sympy import integer_nthroot, divisors
def A363052_gen(startvalue=1): # generator of terms >= startvalue
    for m in count(max(startvalue,1)):
        for k in divisors(m**2,generator=True):
            p, q = integer_nthroot(k**4+(k*m**2<<2),2)
            if q:
                a, b = divmod(p-k**2,k<<1)
                if a > 0 and not b:
                    yield m
                    break
A363052_list = list(islice(A363052_gen(),20)) # Chai Wah Wu, Jul 03 2023