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    $\begingroup$ Or perhaps from John's estate. legacy.com/obituaries/daily-chronicle/… $\endgroup$ Commented Jul 27, 2016 at 22:57
  • $\begingroup$ Thank you for running the experiment. Hagedorn has some tables in his 2009 paper which would give lengths just using prime moduli. Perhaps you could calculate some lower bounds for m(n) for larger n by "filling in" around some of those long intervals? My guess is (for n = 19 where he uses the 19 smallest odd primes to get a covered interval of length 86 =w(n)) that near there you can get m(34) to be 3*w(19) or better. Since Hagedorn has located the intervals covered by prime moduli, extending these should not take much time. Gerhard "Much Computer Time, That Is" Paseman, 2016.07.28. $\endgroup$ Commented Jul 28, 2016 at 20:13
  • $\begingroup$ I think having small composite moduli available changes the problem a lot, so the results for prime moduli are not going to be very useful in practice. $\endgroup$ Commented Jul 28, 2016 at 20:33
  • $\begingroup$ @RobertIsrael, "EDIT: It is now in the OEIS as sequence A275489." -- good job!!! :) $\endgroup$ Commented Aug 27, 2016 at 4:12