A391733 Numbers in whose binary expansion there is exactly one run of 1-bits with an odd length, while all other run of 1-bits are of an even length (or absent).

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 22, 25, 26, 28, 31, 32, 35, 38, 44, 47, 49, 50, 52, 55, 56, 59, 61, 62, 64, 67, 70, 76, 79, 88, 91, 94, 97, 98, 100, 103, 104, 107, 109, 110, 112, 115, 118, 121, 122, 124, 127, 128, 131, 134, 140, 143, 152, 155, 158, 176, 179, 182, 188, 191, 193, 194, 196, 199, 200, 203, 205, 206

Offset

1,2

Comments

Numbers k such that A005940(1+k) is in A229125.

If n is a term, then 2*n is also a term, and vice versa.

Links

Antti Karttunen, Table of n, a(n) for n = 1..13420; all terms <= 2^17

Index entries for sequences related to binary expansion of n.

Formula

{k such that A358769(A005940(1+k)) = 1}.

Example

4, in binary A007088(4) = 100, has one run of 1-bits, with length 1 (which is an odd number), therefore 4 is included in this sequence.
103, in binary 1100111, has one run of two 1-bits and one run of three 1-bits, therefore 103 is included as a term, as there is only one run with an odd length.
107, in binary 1101011, has two runs of two 1-bits and only one run of single 1-bit, thus only one run of 1's with an odd length, therefore 107 is included as a term.

Mathematica

Select[Range[210], Count[Length/@Select[Split[IntegerDigits[#, 2]], #[[1]]==1&], _?OddQ]==1&] (* Harvey P. Dale, May 24 2026 *)

Prog

(PARI)
A005940(n) = { my(p=2, t=1); n--; until(!n\=2, if((n%2), (t*=p), p=nextprime(p+1))); t };
A358769(n) = isprime(core(n));
is_A391733(k) = A358769(A005940(1+k));

Crossrefs

Cf. A005940, A007088, A229125, A358769.

Subsequences: A000079, (at least) the even terms of A000396\{6}, A000668, A233868.

Sequence in context: A080308 A089559 A165741 * A238989 A093696 A190851

Adjacent sequences: A391730 A391731 A391732 * A391734 A391735 A391736

Keyword

nonn,base,easy

Author

Antti Karttunen, Dec 29 2025

Status

approved