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Old 2019-02-15, 20:55   #23
"Dana Jacobsen"
Feb 2011
Bangkok, TH

16068 Posts

Originally Posted by Bobby Jacobs View Post
Is there any way to find out if a number is lucky without testing all of the numbers up to it?
Yes, quite a bit faster, albeit I don't know anything as simple as modern primality testing. It still has a terrible growth rate so gets quite slow if you're spoiled by how fast we can do primality tests.

You can view it as trial division up to the lucky number count of n (approximately n/log(n)). We generate lucky numbers only up to that (at most).

For a concrete example take n = 2^23 = 8388608 and call is_lucky(n+x) for x = 1..100. There are 8 lucky numbers in the range. Only 14 of the 100 values require anything beyond looking at the first 48 lucky numbers. After that we could sieve to ~ 520000. Along the way check if the number we're interested in would be removed.

Times on a Macbook in milliseconds:

2300 Time to generate lucky numbers to 2^23
12 Time to generate lucky numbers to 520000

160 Time for is_lucky(2^23+x) for x=1..100
120 Time for nth_lucky(517138+x) for x=1..8

These are measuring 100 and 8 (respectively) independent calls. If we really wanted the numbers in the sequential range we could just sieve once (to the lucky count of the max value needed). But that isn't the "is_lucky" predicate, which is usually thought of as a single call with an arbitrary integer input.

Last fiddled with by danaj on 2019-02-15 at 21:01
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Old 2019-08-23, 20:45   #24
Bobby Jacobs
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May 2018

2728 Posts

When are we going to try to find the next maximal lucky gap?
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