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%I #18 Oct 31 2020 18:30:40
%S 8,17,24,34,35,40,49,56,64,68,69,70,71,72,81,88,98,99,104,113,120,129,
%T 136,137,138,139,140,141,142,143,145,152,162,163,168,177,184,192,196,
%U 197,198,199,200,209,216,226,227,232,241,248,258,259,264,272,273,274
%N Numbers n such that n in binary representation has a block of exactly a nontrivial triangular number number of zeros.
%C a(n) is the index of zeros in the complement of the triangular number 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 triangular number length >1; otherwise b(n) = 0. The sequence b(n) = 0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,1,... is not yet in the OEIS and is too sparse to be attractively shown.
%D J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 157.
%H Harvey P. Dale, <a href="/A110474/b110474.txt">Table of n, a(n) for n = 1..1000</a>
%H 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.
%F a(n) is in this sequence iff a(n) (base 2) has a block (not a sub-block) of A000217(k) zeros for some k>1.
%e a(1) = 8 because 8 (base 2) = 1000, which has a block of 3 zeros, where 3 is a nontrivial triangular number (A000217(2)).
%e 16 is not an element of this sequence because 16 (base 2) = 10000 which has a block of 4 zeros, which is not a triangular number (even though it has sub-blocks of the triangular number 3 zeros).
%e a(2) = 17 because 17 (base 2) = 10001, which has a block of 3 zeros (and is a Fermat prime).
%e a(4) = 34 because 34 (base 2) = 100010, which has a block of 3 zeros.
%e a(9) = 64 because 64 (base 2) = 1000000, which has a block of 6 zeros, where 6 is a nontrivial triangular number (A000217(3)).
%e 2049 is in this sequence because 2049 (base 2) = 100000000001, which has a block of 10 zeros, where 10 is a nontrivial triangular number (A000217(4)).
%e 65537 is in this sequence because 65537 (base 2) = 10000000000000001, which has a block of 15 zeros, where 15 is a nontrivial triangular number (A000217(5)) and happens to be a Fermat prime.
%e 4194305 is in this sequence because, base 2, has a block of 21 zeros, where 21 is a nontrivial triangular number (A000217(6)),
%t f[n_] := If[Or @@ (First[ # ] == 0 && Length[ # ] > 1 && IntegerQ[(1 + 8*Length[ # ])^(1/2)] &) /@ Split[IntegerDigits[n, 2]], 0, 1]; Select[Range[500], f[ # ] == 0 &] (* _Ray Chandler_, Sep 16 2005 *)
%t ntnQ[n_]:=AnyTrue[Length/@Select[Split[IntegerDigits[n,2]],FreeQ[#,1]&],#>1 && OddQ[ Sqrt[8#+1]]&]; Select[Range[300],ntnQ] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Oct 31 2020 *)
%Y Cf. A000217, A037011, A086747, A110471, A110472.
%K base,easy,nonn
%O 1,1
%A _Jonathan Vos Post_, Sep 08 2005
%E Corrected by _Ray Chandler_, Sep 16 2005
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