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I am having trouble representing this following list of numbers mathematically ,

$$ A_{n1}= (2*3) $$ $$=\color{red}{6}$$ $$ A_{n2} = (2*3)\, , (2*4)\, , (3*4)\, , (2*3*4) $$ $$= \color{red}{6,8,12,24} $$ $$ A_{n3} = (2*3),(2*4),(2*5),(3*4),(3*5),(4*5),(2*3*4),(2*3*5),(2*4*5),(3*4*5),(2*3*4*5) $$ $$ = \color{red}{6,8,10,12,15,20,24,30,40,60,120} $$ $$ A_{n4} = \begin{matrix} (2*3)&(2*3*4)&(2*3*4*5)&(2*3*4*5*6)\\ (2*4)&(2*3*5)&(2*3*4*6)\\ (2*5)&(2*3*6)&(2*3*5*6)\\ (2*6)&(2*4*5)&(2*4*5*6)\\ (3*4)&(2*4*6)&(3*4*5*6)\\ (3*5)&(2*5*6)&\\ (3*6)&(3*4*5)&\\ (4*5)&(3*4*6)&\\ (4*6)&(3*5*6)&\\ (5*6)&(4*5*6)&\\ \end{matrix}$$

$$ =\color{red}{6,8,10,12,12,15,18,20,24,24,30,30,36,40,48,60,60,72,90,120,120,144,180,240,360,720}$$

**Please note that the $A{n4}$ is not a matrix even though it resembles one (written as one) , i did it this way so that you could see where the terms are coming from.

I have read on combinatorics , binomial theorem and others and still am having trouble.

My hopes are to represent each $A_{n}$ mathematically either as a sum , product or any kind of function where i can plug in a value and obtain a term perhaps such as :

$$ F(A_{nK},T)=F(A_{2},3) = (3*4) = 12 $$

This has stumped me for a few days.

I thank you kindly for your help and time with this problem.

No-one Important
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    I dont think there will be a general form for this as you are mentioning in the end, because series is fluctuating because of change in number of terms numbers being multiplied – jeea Apr 01 '20 at 19:33
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    I'm not sure if there is a nice way to write this down as a "closed formula". You are better off computing as-needed, or pre-computing and looking up as-needed, these values. See Algorithm wanted: Enumerate all subsets of a set in order of increasing sums, but instead of increasing sums, you want to have increasing products. – Vepir Apr 29 '20 at 13:13
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    If you are just intersted in stating this mathematically: Let $\mathbb Z_{k}={0,1,2,\dots,k-1}$. Then we can observe the power set of this: Let $\mathcal P_m(\mathbb S)$ be the set of all subsets of some set $\mathbb S$, such that every subset has at least $m$ elements. Define the following multiset: (a set but duplicates are allowed) $$P_k=\left{\prod_{s\in S}(s+2):S\in\mathcal P_2(\mathbb Z_{k+1}) \right} $$ Define $F_k(n)$ as "function that returns the $n$th element of $P_k$, sorted by size" ? – Vepir Apr 29 '20 at 13:30

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