A231204 If n = Sum_{i=0..m} c(i)*2^i, c(i) = 0 or 1, then a(n) = Sum_{i=0..m} (m-i)*c(i).

0, 0, 0, 1, 0, 2, 1, 3, 0, 3, 2, 5, 1, 4, 3, 6, 0, 4, 3, 7, 2, 6, 5, 9, 1, 5, 4, 8, 3, 7, 6, 10, 0, 5, 4, 9, 3, 8, 7, 12, 2, 7, 6, 11, 5, 10, 9, 14, 1, 6, 5, 10, 4, 9, 8, 13, 3, 8, 7, 12, 6, 11, 10, 15, 0, 6, 5, 11, 4, 10, 9, 15, 3, 9, 8, 14, 7, 13, 12, 18, 2, 8, 7, 13, 6, 12, 11, 17, 5, 11, 10, 16, 9, 15, 14, 20, 1, 7, 6, 12

Offset: 0

Author

Jon Perry, Nov 05 2013

Keywords

• nonn
• base
• easy

Comments

A literal interpretation of the binary numbers.

Sum of the number of digits to the left (exclusive) of each 1 in the binary expansion of n. - Gus Wiseman, Jan 09 2023

Examples

			For n=13 we have 1101, so we add 0+1+3=4, getting a(13)=4.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 0..8192

Crossrefs

Cf. A000120, A029931, A230877, A384868

Cf. A000523.

Programs

JavaScript

for (i=0;i<100;i++) {
s=i.toString(2);
o=0;
sl=s.length;
for (j=0;j
				

Maple

f:=proc(n) local t1,m,i;
t1:=convert(n,base,2);
m:=nops(t1)-1;
add((m-i)*t1[i+1], i=0..m);
end; # N. J. A. Sloane, Nov 08 2013

Mathematica

Table[Total[Join@@Position[IntegerDigits[n,2],1]-1],{n,0,100}] (* Gus Wiseman, Jan 09 2023 *)

PARI

a(n) = { my (b=binary(n)); sum(k=1, #b, b[k]*(k-1)) } \\ Rémy Sigrist, Jun 25 2021

Python

def A230204(n): return sum(i for i, j in enumerate(bin(n)[2:]) if j=='1') # Chai Wah Wu, Jan 09 2023

Formula

a(n) = A230877(n) - A000120(n). - Gus Wiseman, Jan 09 2023

Extensions

Edited by N. J. A. Sloane, Nov 08 2013

Name edited by Gus Wiseman, Jan 09 2023