cp's OEIS Frontend

A160530 Positive integers that contain only odd-length runs of 0's and 1's in their binary expansion.

%I A160530 #32 Jan 04 2021 04:08:56
%S A160530 1,2,5,7,8,10,14,17,21,23,29,31,32,34,40,42,46,56,58,62,65,69,71,81,
%T A160530 85,87,93,95,113,117,119,125,127,128,130,136,138,142,160,162,168,170,
%U A160530 174,184,186,190,224,226,232,234,238,248,250,254,257,261,263,273,277,279
%N A160530 Positive integers that contain only odd-length runs of 0's and 1's in their binary expansion.
%C A160530 Let the binary representation of n be thought of as a string of 0's and 1's. By a "run" of 0's or 1's, it is meant either a contiguous substring all of 0's bounded by 1's or the by the edge of the string, or a contiguous substring all of 1's bounded by 0's or the by the edge of the string.
%C A160530 Also, the indices of the compositions that have only odd parts. For the definition of the index of a composition see A298644. For example, 263 is in the sequence since its binary form is 100000111 and the composition [1,5,3] has only odd parts. 132 is not in the sequence since its binary form is 10000100 and the composition [1,4,1,2] also has even parts. The command c(n) from the Maple program yields the composition having index n. - _Emeric Deutsch_, Jan 26 2018
%C A160530 From _Robert Israel_, Jan 26 2018: (Start)
%C A160530 An even number n is in the sequence if and only if n = 2^k*m where k is odd and m is an odd number in the sequence.
%C A160530 An odd number n is in the sequence if and only if n = 2^k*(m+1)-1 where k is odd and m is 0 or an even number in the sequence. (End)
%H A160530 Ivan Neretin, <a href="/A160530/b160530.txt">Table of n, a(n) for n = 1..10000</a>
%p A160530 Runs := proc (L) local j, r, i, k: j := 1; r[j] := L[1]: for i from 2 to nops(L) do if L[i] = L[i-1] then r[j] := r[j], L[i] else j := j+1: r[j] := L[i] end if end do: [seq([r[k]], k = 1 .. j)] end proc: RunLengths := proc (L) map(nops, Runs(L)) end proc: c := proc (n) ListTools:-Reverse(convert(n, base, 2)): RunLengths(%) end proc: A := {}: for n to 280 do if type(product(c(n)[j], j = 1 .. nops(c(n))), odd) = true then A := `union`(A, {n}) else  end if end do: A; # most of the Maple program is due to _W. Edwin Clark._ - _Emeric Deutsch_, Jan 26 2018
%p A160530 # Alternative:
%p A160530 filter:= proc(n) option remember; local t;
%p A160530 if n::even then
%p A160530   t:= padic:-ordp(n,2);
%p A160530   if t::even then return false fi;
%p A160530   procname(n/2^t)
%p A160530 else
%p A160530   t:= padic:-ordp(n+1,2);
%p A160530   if t::even then return false fi;
%p A160530   procname((n+1)/2^t-1)
%p A160530   fi
%p A160530 end proc:
%p A160530 filter(0):= true:
%p A160530 select(filter, [$1..1000]); # _Robert Israel_, Jan 26 2018
%t A160530 Select[Range[300],And@@OddQ/@Length/@Split[IntegerDigits[ #,2]]&] (* _Ray Chandler_, May 19 2009 *)
%o A160530 (Python)
%o A160530 from itertools import groupby
%o A160530 def ok(n): return all(len(list(g))%2 == 1 for k, g in groupby(bin(n)[2:]))
%o A160530 print([i for i in range(1, 280) if ok(i)]) # _Michael S. Branicky_, Jan 04 2021
%Y A160530 Cf. A001196, A160531, A298644.
%K A160530 base,nonn
%O A160530 1,2
%A A160530 _Leroy Quet_, May 17 2009
%E A160530 Extended by _Ray Chandler_, May 19 2009