It's all about the sum of the digits
Pyth, 12 bytes

f}hQ_XsjT;H1


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f: Count up from 1 to find the first number such that:
X ... H1: Increment or insert the value 1 into the dictionary H at index
sjT;: sum of base ten digits of current number
_: values of dictionary
}hQ: check whether input + 1 is contained within.


The first time this is true, exactly input numbers must have the same digit sum as the current number, for the first time.