A361079 - OEIS

%I #30 Mar 16 2023 04:52:41

%S 0,1,1,2,1,3,3,3,1,4,4,5,4,6,6,4,1,5,5,8,5,9,9,7,5,10,10,9,10,10,10,5,

%T 1,6,6,12,6,13,13,13,6,14,14,15,14,16,16,9,6,15,15,18,15,19,19,12,15,

%U 20,20,14,20,15,15,6,1,7,7,17,7,18,18,23,7,19,19

%N Number of integers in [n .. 2n-1] having the same binary weight as n.

%C Or number of steps it takes to double n, where each step goes to the next larger integer with the same binary weight.

%H Alois P. Heinz, <a href="/A361079/b361079.txt">Table of n, a(n) for n = 0..20000</a>

%F a(n) = |{ k in [n, 2n-1] : A000120(k) = A000120(n) }|.

%F ((x-> A057168(x))^a(n))(n) = 2*n.

%F a(n) = A068076(2n) - A068076(n) = A263017(2n) - A263017(n).

%F a(n) = 1 <=> n in { A000079 }.

%e a(9) = 4: 9 -> 10 -> 12 -> 17 -> 18 or in binary: 1001_2 -> 1010_2 -> 1100_2 -> 10001_2 -> 10010_2.

%p b:= proc(n) option remember; uses Bits; local c, l, k;

%p       c, l:= 0, [Split(n)[], 0];

%p       for k while l[k]<>1 or l[k+1]<>0 do c:=c+l[k] od;

%p       Join([1$c, 0$k-c, 1, l[k+2..-1][]])

%p     end:

%p a:= proc(n) option remember; local i, m, t; m, t:=n, 2*n;

%p       for i from 0 while m<>t do m:= b(m) od; i

%p     end:

%p seq(a(n), n=0..100);

%t f[n_] := f[n] = DigitCount[n, 2, 1]; a[n_] := Count[ Array[f, n, n], f[n]]; Array[a, 75, 0] (* _Robert G. Wilson v_, Mar 15 2023 *)

%o (Python)

%o from math import comb

%o def A361079(n):

%o     c, k = 0, 1

%o     for i,j in enumerate(bin(n)[-1:1:-1]):

%o         if j == '1':

%o             c += comb(i+1,k)-comb(i,k)

%o             k += 1

%o     return c # _Chai Wah Wu_, Mar 01 2023

%o (PARI) a(n) = my(w=hammingweight(n)); sum(k=n, 2*n-1, hammingweight(k) == w); \\ _Michel Marcus_, Mar 16 2023

%Y Cf. A000079, A000120, A057168, A068076, A263017.

%K nonn,look,base

%O 0,4

%A _Alois P. Heinz_, Mar 01 2023