This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
a(1) = 4 because there are four primitive solutions (a, b, c, d) as follows: (1, 2, 2, 3), (2, 3, 6, 7), (1, 4, 8, 9), (4, 4, 7, 9) with d < 10.
For n=1, we have the following, where r = (x*y) mod (x+y). (In the last four columns, each number marked by an asterisk is a square.) . r z x y x*y x+y x*y*z x^2+y^2+z^2 - - - - --- --- ----- ----------- 0 1 2 2 4* 4* 4* 9* (solution) 2 1 2 4 8 6 8 21 4 1 2 6 12 8 12 41 6 1 2 8 16* 10 16* 69 3 1 3 3 9 6 9* 19 0 2 3 6 18* 9* 36* 49* (solution) 3 2 3 9 27 12 54 94 0 2 4 4 16* 8 32 36* (solution) 8 2 4 8 32 12 64* 84 5 2 5 5 25* 10 50 54 0 3 6 6 36* 12 108 81* (solution) 7 3 7 7 49* 14 147 107 0 4 8 8 64* 16* 256* 144* (solution) 9 4 9 9 81* 18 324* 178 . z = 2 has the largest number of solutions, so a(1) = 2. The number of solutions for the resulting z cannot exceed A018892(z).
a(n)=my(bc=0,bk=0,L=10^n);for(k=1,L-1,my(c=0,k2=k^2);for(d=max(1,k2\(L-k)+1),k,if(k2%d==0,c++););if(c>bc,bc=c;bk=k););return(bk); \\ Darío Clavijo, Mar 03 2025
def a(n):
# k=x*y and d=x+y
bc, bk, L = 0, None, 10**n
for k in range(1, L):
c, k2 = 0, k * k
for d in range(max(1, k2 // (L - k) + 1), k + 1):
if k2 % d == 0: c += 1
if c > bc:
bc, bk = c, k
return bk
For n=1 the a(1)=1 solution is (1,2,2,3). For n=2 the a(2)=3 solutions are (1,4,8,9), (2,3,6,7) and (2,4,4,6). For n=3 the a(3)=6 solutions are (1,6,18,19), (2,5,14,15), (2,6,9,11), (3,4,12,13), (3,6,6,9) and (4,4,7,9).
n=50;
div={};suc={};A={};
Do[A=Join[A,{Range[1,(1+1/Sqrt[3])q]}],{q,1,n}];
Do[suc=Join[suc,{Length[div]}];div={};For [i=1,i<=Length[Extract[A,q]],i++,div=Join[div,Intersection[Divisors[q^2+(Extract[Extract[A,q],i]-q)^2],Range[2(Extract[Extract[A,q],i]-q),Sqrt[q^2+(Extract[Extract[A,q],i]-q)^2]]]]],{q,1,n}];suc=Rest[Join[suc,{Length[div]}]];matriz={{"q"," ","cuaternas"}};For[j=1,j<=n,j++,matriz=Join[matriz,{{j," ",Extract[suc,j]}}]];MatrixForm[Transpose[matriz]]
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