Thursday, March 31, 2016

Cyclopean primes

After watching the Numberphile Glitch Primes and Cyclops Numbers video yesterday, I thought it a bit of a waste (because of an overly narrow definition) that there should only be one binary Cyclops number that is prime. What if we allow the solitary zero to be anywhere (except of course leftmost) in a binary all-ones number? Well, we would have the terms in Antti Karttunen's A095078, which I'm now calling Cyclopean primes. It would be easy to generate ten or twenty thousand of these and call it a day, but I'm interested in — given that these numbers are of the form 2^n -2^m -1 with m< n-1 — how many primes (values of m) there are for any given n. So, my table starts like this (indices n, bracketed m's):

 2   {0}
 3   {1}
 4   {2,1}
 5   {3,1}
 6   {4,2,1}
 8   {6,5,4,2}
 9   {7,5,3,1}
10   {5,2,1}
11   {3}

There are 38218 solutions up to (and including) n=10000, so an average of 3.8 solutions for a given n. The so-far largest number of solutions is 20, occurring at n=2850 and n=9510.

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