MathLabs

Problem 3

Determine all positive integers kk for which there exist a positive integer mm and a set SS of positive integers such that any integer n>mn>m can be written as a sum of distinct elements of SS in exactly kk ways.
Step 1 of 5: Answer
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k=2a,a=0,1,2,…k=2^a,\quad a=0,1,2,\ldots
Detailed analysis

The valid values are exactly the powers of two, k=2ak=2^a for integers a≥0a\ge0.