cp's OEIS Frontend

A131323 Odd numbers whose binary expansion ends in an even number of 1's.

%I A131323 #49 Apr 22 2025 04:00:38
%S A131323 3,11,15,19,27,35,43,47,51,59,63,67,75,79,83,91,99,107,111,115,123,
%T A131323 131,139,143,147,155,163,171,175,179,187,191,195,203,207,211,219,227,
%U A131323 235,239,243,251,255,259,267,271,275,283,291,299,303,307,315,319,323,331
%N A131323 Odd numbers whose binary expansion ends in an even number of 1's.
%C A131323 Also numbers of the form (4^a)*b - 1 with positive integer a and odd integer b. The sequence has linear growth and the limit of a(n)/n is 6. - _Stefan Steinerberger_, Dec 18 2007
%C A131323 Evil and odious terms alternate. - _Vladimir Shevelev_, Jun 22 2009
%C A131323 Also odd numbers of the form m = (A079523(k)-1)/2. - _Vladimir Shevelev_, Jul 06 2009
%C A131323 As a set, this is the complement of A079523 in the odd numbers. - _Michel Dekking_, Feb 13 2019
%C A131323 From _Ctibor O. Zizka_, Dec 28 2024: (Start)
%C A131323 Numbers k >= 1 such that (k + 1)*(k + 2*r)/2 is not a square for any r >= 1.
%C A131323 Numbers k such that A076114(k + 1) = 0. (End)
%H A131323 Robert Israel, <a href="/A131323/b131323.txt">Table of n, a(n) for n = 1..10000</a>
%H A131323 Thomas Zaslavsky, <a href="/A075326/a075326_2.pdf">Anti-Fibonacci Numbers: A Formula</a>, Sep 26 2016.
%F A131323 a(n) = 2*A079523(n) + 1. - _Michel Dekking_, Feb 13 2019
%e A131323 11 in binary is 1011, which ends with two 1's.
%p A131323 N:= 1000: # to get all terms up to N
%p A131323 Odds:= [seq(2*i+1,i=0..floor((N-1)/2)]:
%p A131323 f:= proc(n) local L,x;
%p A131323    L:= convert(n,base,2);
%p A131323    x:= ListTools:-Search(0,L);
%p A131323    if x = 0 then type(nops(L),even) else type(x,odd) fi
%p A131323 end proc:
%p A131323 A131323:= select(f,Odds); # _Robert Israel_, Apr 02 2014
%t A131323 Select[Range[500], OddQ[ # ] && EvenQ[FactorInteger[ # + 1][[1, 2]]] &] (* _Stefan Steinerberger_, Dec 18 2007 *)
%t A131323 en1Q[n_]:=Module[{ll=Last[Split[IntegerDigits[n,2]]]},Union[ll] =={1} &&EvenQ[Length[ll]]]; Select[Range[1,501,2],en1Q] (* _Harvey P. Dale_, May 18 2011 *)
%o A131323 (PARI) is(n)=n%2 && valuation(n+1,2)%2==0 \\ _Charles R Greathouse IV_, Aug 20 2013
%o A131323 (Python)
%o A131323 from itertools import count, islice
%o A131323 def A131323_gen(startvalue=3): # generator of terms >= startvalue
%o A131323     return map(lambda n:(n<<1)+1,filter(lambda n:(~(n+1)&n).bit_length()&1,count(max(startvalue>>1,1))))
%o A131323 A131323_list = list(islice(A131323_gen(),30)) # _Chai Wah Wu_, Sep 11 2024
%o A131323 (Python)
%o A131323 def A131323(n):
%o A131323     def bisection(f,kmin=0,kmax=1):
%o A131323         while f(kmax) > kmax: kmax <<= 1
%o A131323         kmin = kmax >> 1
%o A131323         while kmax-kmin > 1:
%o A131323             kmid = kmax+kmin>>1
%o A131323             if f(kmid) <= kmid:
%o A131323                 kmax = kmid
%o A131323             else:
%o A131323                 kmin = kmid
%o A131323         return kmax
%o A131323     def f(x):
%o A131323         c, s = n+x, bin(x+1)[2:]
%o A131323         l = len(s)
%o A131323         for i in range(l&1,l,2):
%o A131323             c -= int(s[i])+int('0'+s[:i],2)
%o A131323         return c
%o A131323     return bisection(f,n,n)<<1|1 # _Chai Wah Wu_, Jan 29 2025
%Y A131323 Cf. A076114, A079523, A121539.
%K A131323 nonn,easy
%O A131323 1,1
%A A131323 _Nadia Heninger_ and _N. J. A. Sloane_, Dec 16 2007
%E A131323 More terms from _Stefan Steinerberger_, Dec 18 2007