A037870 a(n) = (1/2)*Sum{|d(i)-e(i)|}, where Sum{d(i)*2^i} is base 2 representation of n and e(i) are digits d(i) in nonincreasing order, for i=0,1,...,m.

0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 1, 2, 1, 1, 0, 1, 1, 1, 1, 2, 1, 1, 1, 2, 2, 2, 1, 2, 1, 1, 1, 2, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• base

Comments

a(n) is also the minimum number of bit swap operations that needs to be performed to pack all the bits of n to the right. - Philippe Beaudoin, Aug 20 2014

a(n) = 0 if A243109(n) = n; 1 + a(A243109(n)) otherwise. - Philippe Beaudoin, Aug 20 2014

The index of the k-th record of this sequence is A020522(k) and corresponds to k ones followed by k zeros. - Philippe Beaudoin, Aug 20 2014

Links

• Philippe Beaudoin, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A007088, A020522, A243109.

Programs

Maple

a:= proc(n) local L,m;
L:= convert(n,base,2);
m:= convert(L,`+`);
m - convert(L[1..m],`+`);
end proc:
seq(a(n),n=1..100); # Robert Israel, Aug 20 2014

Mathematica

A037870[n_] := Total[Abs[# - Sort[#]]/2] & [IntegerDigits[n, 2]];
Array[A037870, 100] (* Paolo Xausa, Mar 07 2025 *)

PARI

a(n) = {d = binary(n); e = vecsort(d); sum(i=1, #d, abs(d[i]-e[i]))/2;} \\ Michel Marcus, Aug 20 2014

Extensions

Definition swapped with A037879. - R. J. Mathar, Oct 19 2015