jmc

We can count that there are 49 factors in the given product. For $n<1,$ all the factors are negative, so the product is negative.

Then for $1<n<3,$ the factor $n-1$ changes sign, and the product becomes positive. For $3<n<5,$ the product changes sign again, and the product becomes negative, so the inequality holds for $n=4.$

Continuing in this way, we see that the inequality holds for $n=4,$ 8, 16, $\dots ,$ 96. For $n>97,$ all the factors are positive, so the total number of such integers is $\boxed{24}.$