Consider a set of (not necessarily consecutive) prime numbers, $S: = \{ p_1, p_2, \ldots, p_k\}.\ $ For each integer $n,$ for each $1\leq j \leq k,$ let (the function) $u_n(p_j)$ be the greatest nonnegative integer such that ${p_j} ^ {u_n(p_j)}$ divides $n.$
Next, define the function $f_S(n):= \displaystyle\sum_{j=1}^{j=k} u_n(p_j).\ $
For example, if $S = ( 2,7,11)\ $ then $f_S(55000) = f_S\left( 2^3 \times 5^4 \times 7^{0} \times {11}^1 \right) = 3 + 0 + 1 = 4,\ $ because $2,7,11\in S,\ $ but $5\notin S.\ $ Next, $ f_S(1) = f_S\left( 2^0 \times 7^0 \times {11}^0 \right) = 0 + 0 + 0 = 0.\ $ Next, $f_S(3) = f_S\left( 2^0\times 3^1 \times 7^0 \times {11}^0 \right) = 0 + 0 + 0 = 0.$
My question is this:
Given a finite ordered list/tuple of $m$ numbers (these are the outputs of $f_S$), does there exist a set $S$ as described above which gives this exact tuple/list of $m$ numbers as output from $m$ consecutive inputs into $f_S ?$
So for example, if someone gave you the output $12-$tuple: $(1,2,0,1,0,4,0,1,0,2,3,2).$
Then the answer for this would be "yes, we can take $S = \{2,5,7\},$ and the consecutive inputs $(115,116,117,\ldots,126).$ This works because $f_S(115) = 1,\ f_S(116) = 2,\ f_S(117) = 0,\ \ldots,\ \ f_S(125) = 3,\ f_S(126) = 2.$"
The question is, does this work for any given (finite) $m-$tuple/list of outputs: can we always find an $S$ and a list of consecutive inputs that output the given list of outputs through $f_S$?
I imagine this is true, because there are many different sets $S$ we can try, and there are also lots of options for our consecutive integers. It may be difficult to find an answer for "crazy" tuples like $(1000, 1, 0, 0, 10^{50}, 10^{30}, 7, 0, 0, 3 ),$ but I don't see any immediate reason why we should not be able to find an $S$ and some consecutive numbers that make this work, using the Chinese remainder theorem and some other results about prime gaps and prime numbers, or from the "abstract algebra" side of prime numbers, like groups $\mathbb{Z}/p\mathbb{Z}.$
Despite my convoluted way of asking it, I find this to be a very natural question, and I would be surprised if it hasn't been investigated before. If someone wants to edit this question to make it read better - for example if there is more widely established notation already used in the literature for any part of my question - then please feel free to do so.
Context(?): I made the question up. I was investigating the sum of indices of prime factorisations of $n!,$ $n$ an integer, for which I have different questions, and I stumbled upon this question which seems very natural to me but I cannot straight away think of an answer.
