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A263017
n is the a(n)-th positive integer having its binary weight.
11
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
1,2
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
FORMULA
a(n) = 1 + A068076(n). - Antti Karttunen, May 22 2017
EXAMPLE
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.
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
PROG
(Perl) # See Links section.
(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
(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
CROSSREFS
KEYWORD
nonn,look,base
AUTHOR
Paul Tek, Oct 07 2015
STATUS
approved