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Feb 4 at 5:33 comment added domotorp "I also conjecture that each $b(k)$ is used in the summation of $c(m)$ no more than once." Isn't this obvious? In binary form, the first (1-digit) of $k$ determines $n+i$, after which the first 0-digit determines $i$. I guess that the addtional sum involving $j$ even ensures that each $k$ occurs exactly once.
Feb 2 at 12:37 history asked Notamathematician CC BY-SA 4.0