A364202 Integers m which can be written as m = p*q = r*s, with 1 <= r < p < q < s <= m and satisfying (p+q) | (s-r).

6, 21, 24, 30, 40, 52, 54, 60, 72, 84, 96, 105, 120, 126, 150, 154, 160, 165, 180, 186, 189, 204, 208, 210, 216, 240, 270, 273, 288, 294, 300, 301, 312, 322, 330, 336, 342, 357, 360, 378, 384, 414, 420, 456, 468, 480, 486, 504, 525, 540, 546, 550, 594, 600

Offset: 1

Author

Jose Aranda, Jul 13 2023

Keywords

• nonn

Comments

Terms may have multiple solutions p,q,r,s, and each has a least quotient k = (s-r) / (p+q).

Those with k=1 are the congruent numbers (A003273) and others are a more general case.

They all share a simple inter-square characterization. The 4 squares are A = (q-p)^2, B = (p+q)^2, C = ((p+q)*k)^2 and D = (r+s)^2. We have B = A + 4m, C = B*(k^2) and D = C + 4m, where 4m is added exclusively to avoid the use of fractions.

Examples

			21 is a term since 21 = 3*7 = 1*21 which has 3+7 = 10 divides 21-1 = 20 (k=2).
So there are 4 squares, in this case, 16, 100, 400 and 484, which are related by this number. In effect, 4*21=+84 jumps from the first to the second, which, multiplied by k^2, gives the third, where +84 gives the fourth.
		

Links

• Jose Aranda, Table of n, a(n) for n = 1..9999

Crossrefs

Cf. A003273 (congruent numbers).

Programs

PARI

isok(k) = my(d=divisors(k)); if (#d >= 4, for (i=1, #d-1, my(r = d[i], s = k/r); if (rMichel Marcus, Jul 17 2023

Extensions

More terms from Alois P. Heinz, Jul 13 2023