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    <p>It is still somewhat unclear, what you are searching for. Quite frequently number theoretic problems allow huge speedups by deriving mathematical properties that the solutions must satisfiy.</p> <p>If you are indeed searching for the integers that maximize the number of non-witnesses for the MR test (i.e. oeis.org/classic/A141768 that you mention) then it might be possible to use that the number of non-witnesses cannot be larger than phi(n)/4 and that the integers for which have this many non-witnesses are either are the product of two primes of the form </p> <blockquote> <p>(k+1)*(2k+1)</p> </blockquote> <p>or they are Carmichael numbers with 3 prime factors. I'd think above some limit all integers in the sequence have this form and that it is possible to verify this by proving an upper bound for the witnesses of all other integers. E.g. integers with 4 or more factors always have at most phi(n)/8 non-witnesses. Similar results can be derived from you formula for the number of bases for other integers.</p> <p>As for micro-optimizations: Whenever you know that an integer is divisible by some quotient, then it is possible to replace the division by a multiplication with the inverse of the quotient modulo 2^64. And the tests n % q == 0 can be replaced by a test</p> <blockquote> <p>n * inverse_q &lt; max_q,</p> </blockquote> <p>where inverse_q = q^(-1) mod 2^64 and max_q = 2^64 / q. Obviously inverse_q and max_q need to be precomputed, to be efficient, but since you are using a sieve, I assume this should not be an obstacle.</p>
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    1. COI will look very seriously into this. I have not been well these past few days; apologies for appearing to ignore your thoughtful comment.
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    2. CO@Charles I tend to think that running your program to find the values of the sequence A141768 is a waste of time. But I also think that your program could be used to find the average number of bases that pass the Miller-Rabin test for a random k-bit integer. Comparing the results with the paper by Damgård, Landrock, Pomerance, "Average case error estimates for the strong probable prime test" would be interesting. But, of course that is only my own opinion.
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    3. COOf course the search is partially inspired by DLP. But I don't think this method will be much use for checking their figures, since the area of interest is perhaps 300 to 3000 bits, where factoring is difficult and sieving impossible.
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