Relation between numbers

Question

Is there a way to find the relation between numbers? I have 2 groups of numbers.

Group A

5 --> 20
6 --> 32
7 --> 112
8 --> 192
9 --> 576
10 --> 1024

Group B

7 --> 56
8 --> 80
9 --> 432
10 --> 672

Is there a way to find relation between numbers and if there's no way to do that except mentally, what is the relation in the two groups?

Answer 1

Unfortunately, there is often no definitive way to determine a pattern within some given set of numbers, since patterns may be as arbitrary as you like. However, there are some ways of seeing patterns a little more clearly.

By way of example, I'm going to rewrite these relations in a slightly different way. Using that as a hint, try and find the relations yourself. I'll place the answer in a spoiler below.

Group A:

$5\to 2^2\cdot5$

$6\to 2^5$

$7\to 2^4\cdot7$

$8\to 2^6\cdot3$

$9\to 2^6\cdot9$

$10\to 2^{10}$

Group B:

$7\to 2^3\cdot7$

$8\to 2^4\cdot5$

$9\to 2^4\cdot3\cdot9$

$10\to 2^5\cdot3\cdot7$

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Group A:

When $n$ is odd, $n\to n2^{n-2}$. When $n$ is even, $n\to2^{n-3}(n-2)$

Group B:

When $n$ is odd, $n\to 2^{n-6}\cdot n(n-3)$. When $n$ is even, $n\to 2^{n/2}(n-3)(n-7)$