A007950 Binary sieve: delete every 2nd number, then every 4th, 8th, etc.

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

Offset: 1

Author

R. Muller

Keywords

• nonn
• easy

Comments

From Charles T. Le (charlestle(AT)yahoo.com), Mar 22 2004: (Start)

This sequence 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; ... (End)

Certainly this sequence contains infinitely many composite numbers, as it has finite density A048651, while the primes have zero density. - Franklin T. Adams-Watters, Feb 25 2011

References

• F. Smarandache, Properties of Numbers, 1972.

Links

• T. D. Noe, Table of n, a(n) for n = 1..10000
• C. Dumitrescu & V. Seleacu, editors, Some Notions and Questions in Number Theory, Vol. I.
• F. Smarandache, Only Problems, Not Solutions!, 4th ed., 1993, Problem 95.
• Index entries for sequences generated by sieves

Crossrefs

Cf. A007951, A000959, A048651.

Programs

Mathematica

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 *)

Extensions

More terms from Robert G. Wilson v, Sep 14 2006