A056791 Weight of binary expansion of n + length of binary expansion of n.

1, 2, 3, 4, 4, 5, 5, 6, 5, 6, 6, 7, 6, 7, 7, 8, 6, 7, 7, 8, 7, 8, 8, 9, 7, 8, 8, 9, 8, 9, 9, 10, 7, 8, 8, 9, 8, 9, 9, 10, 8, 9, 9, 10, 9, 10, 10, 11, 8, 9, 9, 10, 9, 10, 10, 11, 9, 10, 10, 11, 10, 11, 11, 12, 8, 9, 9, 10, 9, 10, 10, 11, 9, 10, 10, 11, 10, 11, 11, 12, 9, 10, 10, 11, 10, 11, 11

Offset

0,2

Links

Michael De Vlieger, Table of n, a(n) for n = 0..16384

Ralf Stephan, Some divide-and-conquer sequences ...

Ralf Stephan, Table of generating functions

Index to sequences related to the complexity of n

Formula

a(n) = a((n - n mod 2) / (2 - n mod 2)) + 1 for n>0, a(0)=1. - Reinhard Zumkeller, Jul 29 2002

a(2n) = a(n)+1, a(2n+1) = a(n)+2. G.f.: 1 + 1/(1-x) * sum(k>=0, (2t+t^2)/(1+t), t=x^2^k). For n>0, a(n) = 2*A000120(n) + A080791(n) = A000120(n) + A029837(n). - Ralf Stephan, Jun 14 2003

a(n) = A000120(n) + A070939(n). - Ruud H.G. van Tol, Mar 20 2026

Example

12 = 1100 in binary, so a(12) = 2+4 = 6.

Maple

a:= n-> add(1+i, i=convert(n, base, 2)):
seq(a(n), n=0..86);  # Alois P. Heinz, Mar 20 2026

Mathematica

Table[If[n==0, 1, s=IntegerDigits[n, 2]; Total@s+Length@s], {n, 0, 100}] (* Giorgos Kalogeropoulos, Sep 13 2021 *)

Prog

(PARI) a(n) = if (n==0, 1, my(b=binary(n)); vecsum(b) + #b); \\ Michel Marcus, Sep 13 2021
(PARI) a(n)= if(n, 1 + logint(n, 2) + hammingweight(n), 1); \\ Ruud H.G. van Tol, Mar 18 2026
(Python)
def a(n): b = bin(n)[2:]; return b.count('1') + len(b)
print([a(n) for n in range(87)]) # Michael S. Branicky, Sep 13 2021
(Python)
def a(n): return n.bit_count() + n.bit_length() if n else 1
print([a(n) for n in range(87)]) # Michael S. Branicky, Mar 18 2026

Crossrefs

Equals A056792 + 1.

Equals A014701 + 2.

Cf. A061313, A037861.

Cf. A000120, A029837, A070939, A080791.

Sequence in context: A200247 A360746 A374066 * A333262 A218767 A334863

Adjacent sequences: A056788 A056789 A056790 * A056792 A056793 A056794

Keyword

nonn,easy,base

Author

N. J. A. Sloane, Sep 01 2000

Extensions

More terms from James Sellers, Sep 06 2000 and from David W. Wilson, Sep 07 2000

Status

approved