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

%I A110472 #17 May 05 2018 14:39:12
%S A110472 16,33,48,64,66,67,80,97,112,129,132,133,134,135,144,161,176,192,194,
%T A110472 195,208,225,240,258,259,264,265,266,267,268,269,270,271,272,289,304,
%U A110472 320,322,323,336,353,368,385,388,389,390,391,400,417,432,448,450,451
%N A110472 Numbers n such that n in binary representation has a block of exactly a semiprime number of zeros.
%C A110472 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.
%D A110472 J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 157.
%H A110472 Vincenzo Librandi, <a href="/A110472/b110472.txt">Table of n, a(n) for n = 1..1000</a>
%H A110472 J.-P. Allouche, <a href="http://www.mat.univie.ac.at/~slc/s/s30allouche.html">Finite Automata and Arithmetic</a>, Séminaire Lotharingien de Combinatoire, B30c (1993), 23 pp.
%e A110472 a(1) = 16 because 16 (base 2) = 10000, which has a block of 4 zeros, where 4 is a semiprime (A001358(1)).
%e A110472 a(2) = 33 because 33 (base 2) = 100001, which has a block of 4 zeros.
%e A110472 a(3) = 48 because 48 (base 2) = 110000, which has a block of 4 zeros.
%e A110472 a(4) = 64 because 64 (base 2) = 1000000, which has a block of 6 zeros, where 6 is a semiprime (A001358(2)).
%e A110472 512 is in this sequence because 512 (base 2) = 1000000000, which has a block of 9 zeros, where 9 is a semiprime (A001358(3)).
%t A110472 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 *)
%t A110472 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 *)
%Y A110472 Cf. A001358, A037011, A086747, A110471, A110474.
%K A110472 base,easy,nonn
%O A110472 1,1
%A A110472 _Jonathan Vos Post_, Sep 08 2005
%E A110472 Extended by _Ray Chandler_, Sep 16 2005