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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.

%I A360796 #31 Aug 05 2023 22:01:14
%S A360796 7,9,11,13,14,17,17,19,20,25,23,29,26,27,29,37,31,40,34,35,38,46,39,
%T A360796 41,44,43,44,54,47,58,49,51,56,53,54,67,62,59,59,70,62,73,64,65,74,78,
%U A360796 69,71,71,75,74,86,76,77,79,83,92,93,83,103
%N 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.
%C A360796 n^2 + a(n)^2 belongs to A007692.
%C A360796 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.
%C A360796 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.
%C A360796 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.
%H A360796 Giedrius Alkauskas, <a href="https://web.vu.lt/mif/g.alkauskas/math/squares.pdf">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</a>.
%e A360796 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.
%p A360796 a :=proc(n::integer) local found::boolean; local N, SQ, i;
%p A360796 found:=false; N:=n+1; SQ:={};
%p A360796 while not found do SQ:=SQ union {N^2}; N:=N+1;
%p A360796 for i from n+1 to N-1 do
%p A360796 if evalb(N^2+n^2-i^2 in SQ) then found:=true; end if;
%p A360796 end do; end do; N end proc;
%Y A360796 Cf. A360619, A000404, A007692.
%K A360796 nonn
%O A360796 1,1
%A A360796 _Giedrius Alkauskas_, Feb 21 2023