Algorithm to get which values make sum of a given number from array

As you can see, the number are base-2, which means you can easily use shift.

You could try this:

private IEnumerable<int> FindBits(int value)
{
    // check for bits.
    for (int i = 0; i < 32; i++)
    {
        // shift 1 by i
        var bitVal = 1 << i;   // you could use (int)Math.Pow(2, i); instead
        // check if the value contains that bit.
        if ((value & bitVal) == bitVal)
            // yep, it did.
            yield return bitVal;
    }
}

This method will check what bits are set and return them as an ienumerable. (which can be converted to an array of list)

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Usage:

// find the bits.
var res = FindBits(40).ToArray();

// format it using the string.join
var str = $"[{string.Join(",", res)}]";

// present the results
Console.WriteLine(str);

Results in [8,32]

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Extra info:

                          counter
00000001 =   1     = 1 << 0
00000010 =   2     = 1 << 1 
00000100 =   4     = 1 << 2
00001000 =   8     = 1 << 3
00010000 =  16     = 1 << 4
00100000 =  32     = 1 << 5
01000000 =  64     = 1 << 6
10000000 = 128     = 1 << 7

Instead of writing all combinations you make a for loop which does the counter.

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Some extra non-sense:

If you like lambda's, you could replace the FindBits with this:

private Func<int, IEnumerable<int>> FindBits = (int value) => Enumerable
    .Range(0, 31)
    .Select(i => 2 << i).Where(i => (value & i) == i);

But it's better to keep it simpel/readable.