OFFSET
1,1
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.
LINKS
Jose Aranda, Table of n, a(n) for n = 1..9999
EXAMPLE
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.
PROG
(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
CROSSREFS
KEYWORD
nonn
AUTHOR
Jose Aranda, Jul 13 2023
EXTENSIONS
More terms from Alois P. Heinz, Jul 13 2023
STATUS
approved