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.
%I A357714 #13 Dec 11 2022 10:19:37
%S A357714 1,2,3,4,3,5,4,6,5,6,4,8,5,7,7,8,5,9,5,9,8,8,6,12,7,8,8,10,6,12,7,11,
%T A357714 9,9,9,14,7,9,9,13,7,13,8,12,12,10,8,16,9,12,10,12,8,14,10,14,11,11,9,
%U A357714 19,9,11,13,14,11,15,9,13,11,15,9,19,10,12,14,14,12,16,10,18,13
%N A357714 a(n) is the number of equations in the set E_{n,b} := {x+2^b*y=n^b, 2^b*x+3^b*y=n^b, ..., k^b*x+(k+1)^b*y=n^b, ..., n^b*x+(n+1)^b*y=n^b} which admit at least one nonnegative integer solution when b is sufficiently large.
%C A357714 Defining a(n,b) as the number of equations of the set E_{n,b} which admit at least one nonnegative integer solution, it's possible to prove the existence of b_0 such that for all b > b_0, a(n,b) = a(n) whose value does not depend on b anymore.
%C A357714 a(n) is the number of positive integers k such that k(k+1) <= n or k divides n or k+1 divides n.
%F A357714 a(n) = ceiling(sqrt(n) - 3/2) + A000005(n).
%F A357714 a(n) ~ A356770(n)/2 as n->infinity.
%F A357714 a(n) <= A356770(n) for all n >= 1.
%e A357714 a(11) = 4 since for all b >= 29 the number of equations of the set E_{11,b} which admit at least one nonnegative integer solution is exactly equal to 4.
%e A357714 a(4) = 4 since for all b >= 1 the number of equations of the set E_{11,b} which admit at least one nonnegative integer solution is exactly equal to 4.
%t A357714 Table[Ceiling[Sqrt[n] - 3/2] + Length[Divisors[n]], {n, 1, 100}]
%Y A357714 Cf. A000005, A356770.
%K A357714 nonn
%O A357714 1,2
%A A357714 _Luca Onnis_, Oct 10 2022