Here's a hint: 0b1000000100000010000001 == 2113665 and 0b1000100010001000100010001 == 17895697. I'd gladly explain what's going on, but it seems that some people have already done it online, so I'll link to them instead. In short, multiplying by the first value "repeats" the input four times, while the second knocks out all but every fourth bit. Finally, we "cast out 15s".
Online resources: [id://spurperl], below, linked to Bit Twiddling Hacks by Sean Eron Anderson of Stanford which contains this and many other bit hacks. Anderson, in turn, links to A Modulo Based Bit Counting Approach For 64 Bit Systems by Roshan James, an explanation of my solution above (more or less). Simply note the differences in integer size (they have 64 bits, while most of us only have 32) and requirements (our input is at most 7 bits, and they allow 16). I personally was inspired by a section of the HAKMEM, as I linked above (this also has a difference in size --- the PDP-1110 had 36-bit integers --- and requirements --- 9 bit input).
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Excellent! Thanks for the brief explanation and for giving me a sturdy nudge in the right direction.
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