For all primes, the arithmetic derivative is $1$.
Product Rule: $(xy)^\prime=x^\prime y+xy^\prime$ and $0^\prime=1^\prime=0$.
Now, there is also some lesser-known sub-part to the arithmetic derivative called the Arithmetic Product Derivative, $P(n)$. In this case, if $$ n=p_1^{x_1} \cdot p_2^{x_2} \cdot \dotsc \cdot p_k^{x_k}, \text{ the prime decomposition of } n $$ then $$ P()=x_1 \cdot p_1^{x_1−1} \cdot x_2 \cdot p_2^{x_2−1} \cdot \dotsc \cdot x_k \cdot p_k^{x_k−1} $$
Are there are infinitely values of $n$ such that $(n)=(n−1)+1$?