|
%I #22 Jan 03 2020 21:50:43
%S 1,3,5,9,11,13,17,21,25,27,29,33,35,37,43,49,51,53,57,59,65,67,69,73,
%T 75,77,81,85,89,91,97,101,107,109,113,115,117,121,123,129,131,133,137,
%U 139,145,149,153,155,157,161,163,165,171,173,177,179,181,185,187,195,197
%N Binary sieve: delete every 2nd number, then every 4th, 8th, etc.
%C 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).
%C "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, ... .)
%C "Conjectures: there are infinitely many primes that belong to this sequence; also infinitely many composite numbers.
%C "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; ... "
%C Certainly this sequence contains infinitely many composite numbers, as it has finite density A048651, while the primes have zero density.
%D F. Smarandache, Properties of Numbers, 1972.
%H T. D. Noe, <a href="/A007950/b007950.txt">Table of n, a(n) for n = 1..10000</a>
%H C. Dumitrescu & V. Seleacu, editors, <a href="http://www.gallup.unm.edu/~smarandache/SNAQINT.txt">Some Notions and Questions in Number Theory, Vol. I</a>.
%H F. Smarandache, <a href="http://www.gallup.unm.edu/~smarandache/OPNS.pdf">Only Problems, Not Solutions!</a>, 4th ed., 1993, Problem 95.
%H <a href="/index/Si#sieve">Index entries for sequences generated by sieves</a>
%t 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 *)
%Y Cf. A007951, A000959, A048651.
%K nonn
%O 1,2
%A R. Muller
%E More terms from _Robert G. Wilson v_, Sep 14 2006
|
