I have the set of the first $n$ primes $\{2,3,5,\ldots,p_n\}$. There are $n^n$ ways of selecting $n$ numbers from this set. Each combination has a number ($C_k$) associated with it and it is the product of its elements. Order the combinations in increasing order of Ck. One of the combinations is $\{2,3,5,\ldots,p_n\}$. $C_p = 2\cdot 3\cdot 5\cdot \ldots\cdot p_n$. Is there any way to say without much computation what combination occurs just after ($C_i$) and before ($C_j$) $C_p$. If that is not possible any bounds would also be helpful. Of course $C_i \le C_p - 1 < C_p < C_p + 1 \le C_j$. Can we get better bounds?
for example, $\{2,3,5\}$ has $27$ ways of selecting $3$ elements from the set. $C_p = 2\cdot 3\cdot 5 = 30$. The closest combination after $C_p$ is $\{3,3,5\} = 3\cdot 3\cdot 5 = 45 = C_j$. And the closest combination before $C_p$ is $\{3,3,3\} = 3\cdot 3\cdot 3 = 27 = C_i$. Is there any way of finding Ci,Cj without enumerating all the combinations? Is there anyway of saying $C_i$ and $C_j$ cannot occur within a certain distance from $C_p$? They cannot occur within a distance of $1$ from $C_p$ but can we find better bounds?
thanks for your answers and comments.