cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

A003159 Numbers whose binary representation ends in an even number of zeros.

Original entry on oeis.org

1, 3, 4, 5, 7, 9, 11, 12, 13, 15, 16, 17, 19, 20, 21, 23, 25, 27, 28, 29, 31, 33, 35, 36, 37, 39, 41, 43, 44, 45, 47, 48, 49, 51, 52, 53, 55, 57, 59, 60, 61, 63, 64, 65, 67, 68, 69, 71, 73, 75, 76, 77, 79, 80, 81, 83, 84, 85, 87, 89, 91, 92, 93, 95, 97, 99, 100, 101, 103, 105
Offset: 1

Views

Author

N. J. A. Sloane, Simon Plouffe

Keywords

Comments

Fraenkel (2010) called these the "vile" numbers.
Minimal with respect to property that parity of number of 1's in binary expansion alternates.
Minimal with respect to property that sequence is half its complement. [Corrected by Aviezri S. Fraenkel, Jan 29 2010]
If k appears then 2k does not.
Increasing sequence of positive integers k such that A035263(k)=1 (from paper by Allouche et al.). - Emeric Deutsch, Jan 15 2003
a(n) is an odious number (see A000069) for n odd; a(n) is an evil number (see A001969) for n even. - Philippe Deléham, Mar 16 2004
Indices of odd numbers in A007913, in A001511. - Philippe Deléham, Mar 27 2004
Partial sums of A026465. - Paul Barry, Dec 09 2004
A121701(2*a(n)) = A121701(a(n)); A096268(a(n)-1) = 0. - Reinhard Zumkeller, Aug 16 2006
A different permutation of the same terms may be found in A141290 and the accompanying array. - Gary W. Adamson, Jun 14 2008
a(n) = n-th clockwise Tower of Hanoi move; counterclockwise if not in the sequence. - Gary W. Adamson, Jun 14 2008
Indices of terms of Thue-Morse sequence A010060 which are different from the previous term. - Tanya Khovanova, Jan 06 2009
The sequence has the following fractal property. Remove the odd numbers from the sequence, leaving 4,12,16,20,28,36,44,48,52,... Dividing these terms by 4 we get 1,3,4,5,7,9,11,12,..., which is the original sequence back again. - Benoit Cloitre, Apr 06 2010
From Gary W. Adamson, Mar 21 2010: (Start)
A conjectured identity relating to the partition sequence, A000041 as polcoeff p(x); A003159, and its characteristic function A035263: (1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, ...); and A036554 indicating n-th terms with zeros in A035263: (2, 6, 8, 10, 14, 18, 22, ...).
The conjecture states that p(x) = A(x) = A(x^2) when A(x) = polcoeff A174065 = the Euler transform of A035263 = 1/((1-x)*(1-x^3)*(1-x^4)*(1-x^5)*...) = 1 + x + x^2 + 2*x^3 + 3*x^4 + 4*x^5 + ... and the aerated variant = the Euler transform of the complement of A035263: 1/((1-x^2)*(1-x^6)*(1-x^8)*...) = 1 + x^2 + x^4 + 2*x^6 + 3*x^8 + 4*x^10 + ....
(End)
The conjecture above was proved by Jean-Paul Allouche on Dec 21 2013. - Gary W. Adamson, Jan 22 2014
If the lower s-Wythoff sequence of s is s, then s=A003159. (See A184117 for the definition of lower and upper s-Wythoff sequences.) Starting with any nondecreasing sequence s of positive integers, A003159 is the limit when the lower s-Wythoff operation is iterated. For example, starting with s=(1,4,9,16,...)=(n^2), we obtain lower and upper s-Wythoff sequences
a=(1,3,4,5,6,8,9,10,11,12,14,...)=A184427;
b=(2,7,12,21,31,44,58,74,...)=A184428.
Then putting s=a and repeating the operation gives a'=(1,3,4,5,7,9,11,12,14,...), which has the same first eight terms as A003159. - Clark Kimberling, Jan 14 2011

Examples

			1=1, 3=11, 5=101 and 7=111 have no (0 = even) trailing zeros, 4=100 has 2 (= even) trailing zeros in the base-2 representation.
2=10 and 6=110 end in one (=odd number) of trailing zeros in their base-2 representation, therefore are not terms of this sequence. - _M. F. Hasler_, Oct 29 2013

References

  • N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Crossrefs

For the actual binary numbers see A280049.
Indices of even numbers in A007814.
Complement of A036554, also one-half of A036554.
Cf. A001285, A010060, A000041, A174065, A292118.

Programs

  • Haskell
    import Data.List (delete)
a003159 n = a003159_list !! (n-1)
a003159_list = f [1..] where f (x:xs) = x : f (delete  (2*x) xs)
-- Reinhard Zumkeller, Nov 04 2011
  • Maple
    filter:= n -> type(padic:-ordp(n,2),even):
select(filter,[$1..1000]); # Robert Israel, Jul 07 2014
  • Mathematica
    f[n_Integer] := Block[{k = n, c = 0}, While[ EvenQ[k], c++; k /= 2]; c]; Select[ Range[105], EvenQ[ f[ # ]] & ]
Select[Range[150],EvenQ[IntegerExponent[#,2]]&] (* Harvey P. Dale, Oct 19 2011 *)
  • PARI
    a(n)=if(n<2,n>0,n=a(n-1); until(valuation(n,2)%2==0,n++); n)
  • PARI
    is(n)=valuation(n,2)%2==0 \\ Charles R Greathouse IV, Sep 23 2012
  • Python
    from itertools import count, islice
def A003159_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:(n&-n).bit_length()&1,count(max(startvalue,1)))
A003159_list = list(islice(A003159_gen(),30)) # Chai Wah Wu, Jul 11 2022
  • Python
    def A003159(n):
    def f(x):
        c, s = n+x, bin(x)[2:]
        l = len(s)
        for i in range(l&1^1,l,2):
            c -= int(s[i])+int('0'+s[:i],2)
        return c
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Jan 29 2025

Formula

a(0) = 1; for n >= 0, a(n+1) = a(n) + 1 if (a(n) + 1)/2 is not already in the sequence, = a(n) + 2 otherwise.
Limit_{n->oo} a(n)/n = 3/2. - Benoit Cloitre, Jun 13 2002
More precisely, a(n) = 3*n/2 + O(log n). - Charles R Greathouse IV, Sep 23 2012
a(n) = Sum_{k = 1..n} A026465(k). - Benoit Cloitre, May 31 2003
a(n+1) = (if a(n) mod 4 = 3 then A007814(a(n) + 1) mod 2 else a(n) mod 2) + a(n) + 1; a(1) = 1. - Reinhard Zumkeller, Aug 03 2003
a(A003157(n)) is even. - Philippe Deléham, Feb 22 2004
Sequence consists of numbers of the form 4^i*(2*j + 1), i>=0, j>=0. - Jon Perry, Jun 06 2004
G.f.: (1/(1-x)) * Product_{k >= 1} (1 + x^A001045(k)). - Paul Barry, Dec 09 2004
a(1) = 1, a(2) = 3, and for n >= 2 we get a(n+1) = 4 + a(n) + a(n-1) - a(a(n)-n+1) - a(a(n-1)-n+2). - Benoit Cloitre, Apr 08 2010
If A(x) is the counting function for a(n) <= x, then A(2^n) = (2^(n+1) + (-1)^n)/3. - Vladimir Shevelev, Apr 15 2010
a(n) = A121539(n) + 1. - Reinhard Zumkeller, Mar 01 2012
A003159 = { N | A007814(N) is even }. - M. F. Hasler, Oct 29 2013

Extensions

Additional comments from Michael Somos
Edited by M. F. Hasler, Oct 29 2013
Incorrect formula removed by Peter Munn, Dec 04 2020

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.