Is there a closed formula for the number of integer divisors?

Definition. Let R be the set of all real numbers. A closed formula on the set R (or its subset) is a finite (the number of elements does not depend on the value of the argument) combination and/or superposition of arithmetic operations and elementary functions — power, exponential, logarithmic, trigonometric, taking the integer/fractional part, etc.

Statement. Let N be the set of all natural numbers. Let d(n) be a function of the number of all distinct natural divisors of n defined on N. There is no closed formula F(x) on the set of all positive real numbers R+ = {x>0) such that the restriction of F(x) to N coincides with d(n).

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The answer is no, so there exists a closed formula satisfying the requirements. Indeed, Prunescu and Sauras-Altuzarra showed that there is a closed formula for $d(n)$ that is built up from the binary operations $$x+y,\qquad \max(x-y,0)=\frac{\sqrt{(x-y)^2}+x-y}{2},\qquad \lfloor x/y\rfloor,\qquad x^y.$$