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

1,1

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.

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.

Example

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)).

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 *)

Crossrefs

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

Sequence in context: A070591 A181452 A151981 * A110502 A095784 A041502

Adjacent sequences: A110469 A110470 A110471 * A110473 A110474 A110475

Keyword

base,easy,nonn

Author

Jonathan Vos Post, Sep 08 2005

Extensions

Extended by Ray Chandler, Sep 16 2005

Status

approved