A263017 n is the a(n)-th positive integer having its binary weight.

1, 2, 1, 3, 2, 3, 1, 4, 4, 5, 2, 6, 3, 4, 1, 5, 7, 8, 5, 9, 6, 7, 2, 10, 8, 9, 3, 10, 4, 5, 1, 6, 11, 12, 11, 13, 12, 13, 6, 14, 14, 15, 7, 16, 8, 9, 2, 15, 17, 18, 10, 19, 11, 12, 3, 20, 13, 14, 4, 15, 5, 6, 1, 7, 16, 17, 21, 18, 22, 23, 16, 19, 24, 25, 17

Offset: 1

Author

Paul Tek, Oct 07 2015

Keywords

• nonn
• look
• base

Comments

Binary weight is given by A000120.

a(2^k) = k+1 for any k>=0.

a(2^k-1) = 1 for any k>0.

a(A057168(k)) = a(k)+1 for any k>0.

a(A036563(k+1)) = k for any k>0.

Ordinal transform of A000120. - Alois P. Heinz, Dec 23 2018

Examples

			The numbers with binary weight 3 are: 7, 11, 13, 14, 19, ...
Hence: a(7)=1, a(11)=2, a(13)=3, a(14)=4, a(19)=5, ...
And more generally: a(A014311(k))=k for any k>0.
		

Links

• Paul Tek, Table of n, a(n) for n = 1..10000
• Paul Tek, PERL program for this sequence

Crossrefs

One more than A068076.

Cf. A000120, A036563, A057168, A254524, A286478, A286552, A286554, A286558.

Cf. A263109, A263110.

Programs

Haskell

import Data.IntMap (empty, findWithDefault, insert)
a263017 n = a263017_list !! (n-1)
a263017_list = f 1 empty where
   f x m = y : f (x + 1) (insert h (y + 1) m) where
           y = findWithDefault 1 h m
           h = a000120 x
-- Reinhard Zumkeller, Oct 09 2015

Maple

a:= proc() option remember; local a, b, t; b, a:=
      proc() 0 end, proc(n) option remember; a(n-1);
        t:= add(i, i=convert(n, base, 2)); b(t):= b(t)+1
      end; a(0):=0; a
    end():
seq(a(n), n=1..120);  # Alois P. Heinz, Dec 23 2018

Perl

# See Links section.

Python

from math import comb
def A263017(n):
    c, k = 1, 0
    for i,j in enumerate(bin(n)[-1:1:-1]):
        if j == '1':
            k += 1
            c += comb(i,k)
    return c # Chai Wah Wu, Mar 01 2023

Formula

a(n) = 1 + A068076(n). - Antti Karttunen, May 22 2017