A360796 a(n) > n is the smallest integer such that there exist integers n < c <= d < a(n) satisfying n^2 + a(n)^2 = c^2 + d^2.

7, 9, 11, 13, 14, 17, 17, 19, 20, 25, 23, 29, 26, 27, 29, 37, 31, 40, 34, 35, 38, 46, 39, 41, 44, 43, 44, 54, 47, 58, 49, 51, 56, 53, 54, 67, 62, 59, 59, 70, 62, 73, 64, 65, 74, 78, 69, 71, 71, 75, 74, 86, 76, 77, 79, 83, 92, 93, 83, 103

Offset: 1

Author

Giedrius Alkauskas, Feb 21 2023

Keywords

• nonn

Comments

n^2 + a(n)^2 belongs to A007692.

The identity n^2 + (2*n + 5)^2 = (n+4)^2 + (2*n + 3)^2 shows that a(n) <= 2*n + 5. The last case when the equality holds is n = 16.

a(n) = a(n+1) has infinitely many solutions. This holds, in particular, when n = (u*v + u + v - 1) * (u*v - 2)/2 - 1 for positive integers u, v satisfying v+2 <= u <= 6*v - 3.

a(n-1) = a(n) = a(n+1) holds for n = (3*v^2 + 5*v + 1) * (6*v^2 + 3*v - 2), v >= 3.

Examples

			a(10) = 25, since 10^2 + 25^2 = 14^2 + 23^2, and no integers b, c, d exist satisfying 10 < c <= d < b < 25 and 10^2 + b^2 = c^2 + d^2.
		

Links

• Giedrius Alkauskas, On the function which is defined by the minimal solution to n^2 + f(n)^2 = a^2 + b^2, a, b, f(n) > n.

Crossrefs

Cf. A360619, A000404, A007692.

Programs

Maple

a :=proc(n::integer) local found::boolean; local N, SQ, i;
found:=false; N:=n+1; SQ:={};
while not found do SQ:=SQ union {N^2}; N:=N+1;
for i from n+1 to N-1 do
if evalb(N^2+n^2-i^2 in SQ) then found:=true; end if;
end do; end do; N end proc;