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A110474
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Numbers n such that n in binary representation has a block of exactly a nontrivial triangular number number of zeros.
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6
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8, 17, 24, 34, 35, 40, 49, 56, 64, 68, 69, 70, 71, 72, 81, 88, 98, 99, 104, 113, 120, 129, 136, 137, 138, 139, 140, 141, 142, 143, 145, 152, 162, 163, 168, 177, 184, 192, 196, 197, 198, 199, 200, 209, 216, 226, 227, 232, 241, 248, 258, 259, 264, 272, 273, 274
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OFFSET
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1,1
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COMMENTS
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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.
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REFERENCES
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J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 157.
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LINKS
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FORMULA
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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.
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EXAMPLE
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a(1) = 8 because 8 (base 2) = 1000, which has a block of 3 zeros, where 3 is a nontrivial triangular number (A000217(2)).
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).
a(2) = 17 because 17 (base 2) = 10001, which has a block of 3 zeros (and is a Fermat prime).
a(4) = 34 because 34 (base 2) = 100010, which has a block of 3 zeros.
a(9) = 64 because 64 (base 2) = 1000000, which has a block of 6 zeros, where 6 is a nontrivial triangular number (A000217(3)).
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)).
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.
4194305 is in this sequence because, base 2, has a block of 21 zeros, where 21 is a nontrivial triangular number (A000217(6)),
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MATHEMATICA
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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 *)
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 *)
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CROSSREFS
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KEYWORD
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base,easy,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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