A198193 Replace 2^k in the binary representation of n with n+(k-L) where L = floor(log(n)/log(2)).

0, 1, 2, 5, 4, 8, 11, 18, 8, 15, 18, 28, 23, 35, 39, 54, 16, 30, 33, 50, 38, 57, 61, 83, 47, 70, 74, 100, 81, 109, 114, 145, 32, 61, 64, 96, 69, 103, 107, 144, 78, 116, 120, 161, 127, 170, 175, 221, 95, 141, 145, 194, 152, 203, 208, 262, 165, 220, 225, 283

Offset

0,3

Comments

That is, if n = 2^a + 2^b + 2^c + ... then a(n) = (n+(a-L)) + (n+(b-L)) + (n+(c-L)) + ...).

Links

Table of n, a(n) for n=0..59.

Formula

Let L = A000523(n), then a(n) = (n-L)*A000120(n) + A073642(n).

Example

a(4) = (4+(2-2)) = 4 because int(log(4)/log(2)) = 2 and 4 = 2^2.
a(6) = (6+(2-2)) + (6+(1-2)) = 11 because int(log(6)/log(2)) = 2 and 6 = 2^2 + 2^1.

Maple

read("transforms") :
A198193 := proc(n)
        (n-A000523(n))*wt(n)+A073642(n) ;
end proc:
seq(A198193(n), n=0..20) ; # R. J. Mathar, Nov 17 2011

Mathematica

Table[b = Reverse[IntegerDigits[n, 2]]; L = Length[b] - 1; Sum[b[[k]] (n + k - 1 - L), {k, Length[b]}], {n, 0, 59}] (* T. D. Noe, Nov 01 2011 *)

Prog

(MATLAB)
% n is number of terms to be computed, b is the base. The examples all use b=2:
function [V] = revAddition(n, b)
   for i = 0:n
      k = i;
      if (i > 0)
         l = floor(log(i)/log(b));
      end
      s = 0;
      while(k ~= 0)
         if ((i-l) >= 0)
            s = s + mod(k, b)*(i-l);
         end
         l = l - 1;
         k = (k - mod(k, b))/b;
      end
      V(i+1) = s;
   end
end
(Python)
def A198193(n): return sum((n-i)*int(j) for i, j in enumerate(bin(n)[2:])) # Chai Wah Wu, Mar 13 2021

Crossrefs

Cf. A000120, A073642, A198192.

Sequence in context: A183542 A328203 A080031 * A316905 A214533 A065221

Adjacent sequences: A198190 A198191 A198192 * A198194 A198195 A198196

Keyword

nonn,base

Author

Brian Reed, Oct 26 2011

Status

approved