A110474 Numbers n such that n in binary representation has a block of exactly a nontrivial triangular number number of zeros.

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

Offset: 1

Author

Jonathan Vos Post, Sep 08 2005

Keywords

• base
• easy
• nonn

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.

Examples

			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)),
		

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.

Crossrefs

Cf. A000217, A037011, A086747, A110471, A110472.

Programs

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 *)

Extensions

Corrected by Ray Chandler, Sep 16 2005