I have a function that looks like $$f_n(i) = \max\left(\left\lfloor \frac{i-n}{2} \right \rfloor + 1, 0\right)$$

and I would like to write it as a nice arithmetic function. To give an idea, another example where I can find a suitable answer is $$g_n(i) = \max\left(i-n+1, 0\right)$$

where I get $g_n(i+n) = i+1 = \tau(p^{i})$.

But in the case of $f$ it would end with something like $\tau(p^{\lfloor i/2\rfloor})$, and... it does not seem very multiplicative, is it?

  • 2
    $\begingroup$ In order to write $f_n$ as multiplicative function, it should be multiplicative. Is it? $\endgroup$ – Hagen von Eitzen May 26 '17 at 9:44
  • $\begingroup$ What do you understand under nice? $f_n(i)$ is already nice in this form - at least, it is explicit=) $\endgroup$ – TZakrevskiy May 26 '17 at 9:46
  • $\begingroup$ @HagenvonEitzen Indeed it does not seem to be, however maybe is there a good decomposition of it, essentially as multiplicative/sum of multiplicative functions ? (I do not succeeded in trying to split the cases to getting rid of the division as well as the floor part...) $\endgroup$ – Wolker May 26 '17 at 9:52

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