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A007950
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Binary sieve: delete every 2nd number, then every 4th, 8th, etc.
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7
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1, 3, 5, 9, 11, 13, 17, 21, 25, 27, 29, 33, 35, 37, 43, 49, 51, 53, 57, 59, 65, 67, 69, 73, 75, 77, 81, 85, 89, 91, 97, 101, 107, 109, 113, 115, 117, 121, 123, 129, 131, 133, 137, 139, 145, 149, 153, 155, 157, 161, 163, 165, 171, 173, 177, 179, 181, 185, 187, 195, 197
(list;
graph;
refs;
listen;
history;
text;
internal format)
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OFFSET
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1,2
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COMMENTS
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Comments from C. Le (charlestle(AT)yahoo.com), Mar 22 2004: "A007950 and A007951 are particular cases of the Smarandache n-ary sequence sieve (for n=2 and respectively n=3).
"Definition of Smarandache n-ary sieve (n >= 2): Starting to count on the natural numbers set at any step from 1: - delete every n-th numbers; - delete, from the remaining numbers, every (n^2)-th numbers; ... and so on: delete, from the remaining ones, every (n^k)-th numbers, k = 1, 2, 3, ... .)
"Conjectures: there are infinitely many primes that belong to this sequence; also infinitely many composite numbers.
"Smarandache general-sequence sieve: Let u_i > 1, for i = 1, 2, 3, ..., be a strictly increasing positive integer sequence. Then from the natural numbers: - keep one number among 1, 2, 3, ..., u_1 - 1 and delete every u_1 -th numbers; - keep one number among the next u_2 - 1 remaining numbers and delete every u_2 -th numbers; ... and so on, for step k (k >= 1): - keep one number among the next u_k - 1 remaining numbers and delete every u_k -th numbers; ... "
Certainly this sequence contains infinitely many composite numbers, as it has finite density A048651, while the primes have zero density.
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REFERENCES
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F. Smarandache, Properties of Numbers, 1972.
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LINKS
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MATHEMATICA
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t = Range@200; f[n_] := Block[{k = 2^n}, t = Delete[t, Table[{k}, {k, k, Length@t, k}]]]; Do[ f@n, {n, 6}]; t (* Robert G. Wilson v, Sep 14 2006 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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R. Muller
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EXTENSIONS
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STATUS
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approved
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