|
%I #107 Jul 23 2023 06:04:37
%S 6,21,24,30,40,52,54,60,72,84,96,105,120,126,150,154,160,165,180,186,
%T 189,204,208,210,216,240,270,273,288,294,300,301,312,322,330,336,342,
%U 357,360,378,384,414,420,456,468,480,486,504,525,540,546,550,594,600
%N 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).
%C Terms may have multiple solutions p,q,r,s, and each has a least quotient k = (s-r) / (p+q).
%C Those with k=1 are the congruent numbers (A003273) and others are a more general case.
%C 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.
%H Jose Aranda, <a href="/A364202/b364202.txt">Table of n, a(n) for n = 1..9999</a>
%e 21 is a term since 21 = 3*7 = 1*21 which has 3+7 = 10 divides 21-1 = 20 (k=2).
%e 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.
%o (PARI) isok(k) = my(d=divisors(k)); if (#d >= 4, for (i=1, #d-1, my(r = d[i], s = k/r); if (r<s, for (j=2, #d, my(p = d[j], q = k/p);if (p<q, if (!((s-r) % (p+q)), return(1));););););); \\ _Michel Marcus_, Jul 17 2023
%Y Cf. A003273 (congruent numbers).
%K nonn
%O 1,1
%A _Jose Aranda_, Jul 13 2023
%E More terms from _Alois P. Heinz_, Jul 13 2023
|
