Problem 6

Find all positive integers

n

for which there exist nonnegative integers

a_1,\dots,a_n

such that

\sum_{i=1}^n2^{-a_i}=\sum_{i=1}^n i3^{-a_i}=1

Step 4 of 4: Check the five base sequences

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In plain words

The bases cover lengths 1,5,9,13,17; the odd-to-even step then covers the adjacent even lengths.

(0);\ (2,1,3,4,4);\ (2,3,3,3,3,4,4,4,4);\ (2,3,3,4,4,4,5,4,4,5,4,5,5);\ (3,2,2,4,4,5,5,6,5,6,6,6,6,6,6,6,5)

Detailed analysis

Direct substitution verifies feasibility for the exponent lists of lengths $1,5,9,13,17$ displayed in the formula. The odd-to-even step gives lengths $2,6,10,14,18$ , and the 11-point extension gives every later length $4m+13\equiv1\pmod4$ ; applying the odd-to-even step again gives the corresponding $2\pmod4$ lengths. Hence every $n\equiv1,2\pmod4$ is attainable, matching the necessary condition.