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A110472 Numbers n such that n in binary representation has a block of exactly a semiprime number of zeros. 7
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 12 12:20 EDT 2020. Contains 336439 sequences. (Running on oeis4.)