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eliminates this ambiguity. Here's an idea: changing "The third surviving number" to "The next surviving number" will avoid the temptation to think it's third because the previous interval was 3, and therefore that the 7th surviving number will be picked as the interval for the next pass. (As would showing one more pass of the process, for that matter.)
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This claim is easy to find verbatim on the internet, but I can't find any more information, does somebody have a more precise year, or possibly details about the story? "Stanisław Ulam was the first to discuss these numbers, around 1955. He named them "lucky" because of a connection with a story told
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I produced, for lucky numbers, a picture analogous to the Ulam spiral. It appeared to show similar diagonal stripes. However, a larger spiral with lucky numbers to 200,000 looks considerably more random. I think I will not bother to upload the picture when my account is verified in a few days.
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primes" (where I have simply replaced "lucky" with "even"). My statement is clearly not true, since there is only one even prime. The two categories are almost completely disjoint. Just because there are an infinite number of prime numbers and an infinite number of lucky numbers doesn't mean that
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Of course, 1 and 2 are equivalent, since after pass 1 the interval (hence the index of the first number to be eliminated in the pass) is always greater than its index in the list. And 3 doesn't produce the sequence given in the article. But still, it would be better to phrase it in a way that
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which says: "Many asymptotic properties of the prime numbers are shared by the lucky numbers. The asymptotic density is 1/ln N, just as the prime number theorem". So the sum of reciprocals diverges like the prime numbers.
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I agree that the "1st step" does not fit well into the general scheme (among others, because 2 itself is also removed, and because it's the 1st step but the 2nd number in the list, while the 2nd step uses
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But maybe there is indeed a better way to describe it that avoids the incongruity of the first step differing from the remainder. MFH's description seems to have been tweaked from
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OTOH, it seems that a function islucky(n) is more difficult to write than isprime(n) (which needs only checking for divisibility by 2 and then by all odd numbers : -->
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Gardiner, Lazarus, Metropolis and Ulam did not suggest to call the defining sieve "the sieve of
Josephus Flavius". They suggested this name for another sieve.
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It seems like this could be described more clearly. As it is, the even-number-removing step seems out of place. Something like this, maybe?
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Article states at the end: "There are infinitely many lucky numbers. It is not known whether there are also infinitely many lucky primes..."
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the 2nd number from the list). On mathworld, EWW starts out with the odd numbers (which of course could also be considered as odd...)
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I was just noticing this myself. There's an ambiguity in the way it's described. Is the interval by which to count at each pass:
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Is there any hint somewhere about how this could be done "somehow efficiently" (in particular, limiting memory usage)? —
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for n=2 to infinity, do let x = the nth number of the current list; remove every xth number from the current list;
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Do lucky numbers form a large set (just like primes)? If not, is there known any upper bound of sum of reciprocals?
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The article says: "Lucky numbers share some properties with primes, such as asymptotic behaviour according to the
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No, we actually can't tell from that statement. For example, we could also say: "There are infinitely many
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Haven't done maths for 20 years but there must be an infinite number of lucky primes, isn't that right?
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The entire article (including the
References) has been copied word-for-word from the following link -
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on
Knowledge. If you would like to participate, please visit the project page, where you can join
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Go check out the bottom of the page you linked there. Notice removed. --SomeoneWhoCanRead
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L = all positive integers; X = 1 (= min(L)) do forever X = min { y in L | y : -->
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An alternative description could be: start with the smallest ("remaining") number : -->
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Twin lucky numbers and twin primes also appear to occur with similar frequency.
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looked up using the previous interval as an index into the list?
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the next survivor, after the one that was used on the last pass?
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numbers. It is not known whether there are also infinitely many
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1, which is 2. Then use the smallest remaining number : -->
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the 2nd, 3rd, 4th, 5th, etc. of those left at each stage?
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2, which is 3. Then, the smallest remaining number : -->
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