The general philosophy is that multiplication and addition do not "see" each other. So the fact that one knows the multiplicative structure of n does not say anything about the multiplicative structure of n+1. There are several demonstrations of this philosophy.

One of them concerns twin primes: a well-known heuristic, first exploited by Cramer, says that a random number $n$ is prime with probability $1/\log n$ (this is supported by the Prime Number Theorem). However, if we assume that $n$ is a prime number, then it is believed that the probability of $n+2$ being a prime number is still $1/\log n$, provided that there no local obstructions to this (for example, if $n\equiv1\pmod 3$, then this is trivially false). In other words, $n+2$ does not "know" whether $n$ is prime or not. Indeed, a quantitative form of the twin prime conjecture states that

$$|\{n\le x:n~{\rm and}~n+2~{\rm are~prime~numbers}\}|\sim\frac{cx}{\log^2x},$$

where $c$ is some constant which arises due to the local obstructions mentioned above.

A second demonstration of this philosophy is the Erdos-Szemeredi conjecture which, in its simplest form, states that if $A$ is a set of integers and we set

$$A+A=\{a+b:a,b\in A\}\quad{\rm and}\quad A\cdot A=\{a\cdot b:a,b\in A\},$$

then

$$\max\{|A+A|,|A\cdot A|\}\ge c_\epsilon|A|^{2-\epsilon}$$

for every $\epsilon>0$, where $c_\epsilon$ is some constant that depends only on $\epsilon$. Roughly, this conjecture says that $A$ cannot have both additive and multiplicative structure, which would reduce the cardinality of $A+A$ and $A\cdot A$.