On a problem of V. G. Sprindzhuk

We consider estimates of the function
$$ S(t)=\prod_{p\mid t} p $$
equal to the square-free part of the positive integer argument $t$. V. G. Sprindzhuk posed the following problem. Is there a constant $c>0$ such that the inequality
$$ S((n+1)\ldots (n+k))<k^k $$
is fulfilled for an infinite number of pairs of positive integers $n$ and $k$ such that $k<\ln^c n$? We prove that there exist positive constants $c_7,\ldots,c_{10}$ such that for $n\geq c_7$
$$ S((n+1)\ldots (n+k))\geq p_1\ldots p_{s(k)},\quad s(k)=k+[c_8k/\ln(2k)] $$
if $1\leq k\leq c_9\sqrt{n/\ln n}$;
$$ S((n+1)\ldots (n+k))<p_1\ldots p_k $$
if $k\geq c_{10}\sqrt{n/\ln n}$.
In the paper, we obtain several other estimates of the function $S(t)$ and discuss some conjectures concerning $S(t)$ and derive corollaries of those conjectures.