On the many prime number investigation sites across the web I haven't been able to find the answer. Also my math isn't good enough to compute it from first principles.
So, what is the least prime that has 32 1-bits? Of course this refers to its base 2, i.e., binary representation. Programmer-speak for this would be, "32 'set' bits.'
Layperson explanation of the logic behind finding the answer would be an appreciated bonus.
[edit:] Summary of Answers
The answer is $8581545983$. By assuming the answer has just a single 0-bit, we discover that there are $5$ such ($33$-bit) primes with exactly $32$ 1-bits. Failing that, we would have gone on to try two 0-bits, etc.
The subtle point that confused both of the experts (and myself) is that even though we're seeking the lowest value here, it's incorrect to scan the substitution positions in right←to←left order of increasing significance.
The mis-intuition is caused by the fact that each trial replaces a $1$ with a $0$ at the given bit position—and not the other way around—so the higher the bit significance, the greater the reduction of numeric power. If we were to begin evaluating at the LSB (from the bottom of the following list) and proceed leftwards (upwards), we'd be fruitlessly starting at the candidate with the highest (i.e., least-reduced) overall value.
So the correct scan of values instead proceeds by testing 0-bit positions from left→to→right, as follows (from top-to-bottom):
011111111111111111111111111111111 4294967295 value 101111111111111111111111111111111 6442450943 increase 110111111111111111111111111111111 7516192767 ↓ 111011111111111111111111111111111 8053063679 ↓ 111101111111111111111111111111111 8321499135 ↓ 111110111111111111111111111111111 8455716863 ↓ 111111011111111111111111111111111 8522825727 111111101111111111111111111111111 8556380159 111111110111111111111111111111111 8573157375 111111111011111111111111111111111 8581545983 prime ←- answer 111111111101111111111111111111111 8585740287 111111111110111111111111111111111 8587837439 prime 111111111111011111111111111111111 8588886015 111111111111101111111111111111111 8589410303 prime 111111111111110111111111111111111 8589672447 111111111111111011111111111111111 8589803519 111111111111111101111111111111111 8589869055 111111111111111110111111111111111 8589901823 prime 111111111111111111011111111111111 8589918207 111111111111111111101111111111111 8589926399 111111111111111111110111111111111 8589930495 111111111111111111111011111111111 8589932543 111111111111111111111101111111111 8589933567 111111111111111111111110111111111 8589934079 111111111111111111111111011111111 8589934335 111111111111111111111111101111111 8589934463 111111111111111111111111110111111 8589934527 ↑ 111111111111111111111111111011111 8589934559 ↑ 111111111111111111111111111101111 8589934575 ↑ 111111111111111111111111111110111 8589934583 prime ↑ 111111111111111111111111111111011 8589934587 higher 111111111111111111111111111111101 8589934589 0-bit 111111111111111111111111111111110 8589934590 significance
Knowing $2^{32}-1$ isn't prime, I found $2^{33}-1 - 2^{23}=8581545983 $ just by starting with a string of 33 ones and replacing a one with a zero starting from second most significant bit and moving right.
Robert Israel's OEIS link (https://oeis.org/A061712) doesn't list any non-trivial math properties, but this seems pretty trivial to write a program for. Start with a long string of binary ones ($2^k-1$) and try setting 1 one to zero, 2 ones to zero with $2^{k+1}-1$, etc.