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  • $\begingroup$ The sum from $n$ to infinity of the one over the log of $2^n+m$ is infinite, but I can't imagine trying to prove the existence of such primes with current technology. Even getting density 1 seems impossible to me - you would need to look to $n$ exponentially large in $m$. $\endgroup$ Commented Jun 14, 2020 at 23:49
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    $\begingroup$ The sequence "least prime with Hamming distance 1 from the k'th odd integer" starts $3, 2, 7, 3, 11, 3, 5, 7, 19, 3, 5, 7, 17, \ldots$. It doesn't seem to be in the OEIS yet, but should be. Are you interested in contributing it? If you don't wish to, I can (with a link to this question). $\endgroup$ Commented Jun 15, 2020 at 1:55
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    $\begingroup$ Thanks very much for your answer, I wasn't aware of the (dual) Sierpiński numbers. I'll go ahead and add this to the OEIS. $\endgroup$ Commented Jun 15, 2020 at 22:43
  • $\begingroup$ I think this question has already been asked in a slightly different form and answered here: mathoverflow.net/questions/316867/… $\endgroup$ Commented Aug 1, 2021 at 6:17