OFFSET
1,1
COMMENTS
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.
a(n) is in this sequence iff a(n) (base 2) has a block (not a subblock) of A000217(k) zeros for some k>1.
REFERENCES
J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 157.
LINKS
Harvey P. Dale, 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) = 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 subblocks 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)),
MATHEMATICA
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 *)
CROSSREFS
KEYWORD
base,easy,nonn
AUTHOR
Jonathan Vos Post, Sep 08 2005
EXTENSIONS
Corrected by Ray Chandler, Sep 16 2005
STATUS
approved