A110472 Numbers n such that n in binary representation has a block of exactly a semiprime number of zeros.

16, 33, 48, 64, 66, 67, 80, 97, 112, 129, 132, 133, 134, 135, 144, 161, 176, 192, 194, 195, 208, 225, 240, 258, 259, 264, 265, 266, 267, 268, 269, 270, 271, 272, 289, 304, 320, 322, 323, 336, 353, 368, 385, 388, 389, 390, 391, 400, 417, 432, 448, 450, 451

Offset: 1

Author

Jonathan Vos Post, Sep 08 2005

Keywords

• base
• easy
• nonn

Comments

a(n) is the index of zeros in the complement of the semiprime analog of the Baum-Sweet sequence, which is b(n) = 1 if the binary representation of n contains no block of consecutive zeros of exactly semiprime length; otherwise b(n) = 0.

Examples

			a(1) = 16 because 16 (base 2) = 10000, which has a block of 4 zeros, where 4 is a semiprime (A001358(1)).
a(2) = 33 because 33 (base 2) = 100001, which has a block of 4 zeros.
a(3) = 48 because 48 (base 2) = 110000, which has a block of 4 zeros.
a(4) = 64 because 64 (base 2) = 1000000, which has a block of 6 zeros, where 6 is a semiprime (A001358(2)).
512 is in this sequence because 512 (base 2) = 1000000000, which has a block of 9 zeros, where 9 is a semiprime (A001358(3)).
		

References

• J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 157.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• J.-P. Allouche, Finite Automata and Arithmetic, Séminaire Lotharingien de Combinatoire, B30c (1993), 23 pp.

Crossrefs

Cf. A001358, A037011, A086747, A110471, A110474.

Programs

Mathematica

f[n_] := If[Or @@ (First[ # ] == 0 && Plus @@ Last /@ FactorInteger[Length[ # ]] == 2 &) /@ Split[IntegerDigits[n, 2]], 0, 1]; Select[Range[450], f[ # ] == 0 &] (* Ray Chandler, Sep 16 2005 *)
Select[Range[500],AnyTrue[Length/@Select[Split[IntegerDigits[#,2]],#[[1]] == 0&],PrimeOmega[#]==2&]&] (* The program uses the AnyTrue function from Mathematica version 10 *) (* Harvey P. Dale, May 05 2018 *)

Extensions

Extended by Ray Chandler, Sep 16 2005