A056791 - OEIS

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

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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

EXAMPLE

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

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

(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

CROSSREFS

Equals A056792 + 1.

Equals A014701 + 2.