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A290090 a(n) is the number of proper divisors of n that are odious (A000069).

%I A290090 #17 Oct 04 2017 14:04:02
%S A290090 0,1,1,2,1,2,1,3,1,2,1,3,1,3,1,4,1,2,1,3,2,3,1,4,1,3,1,5,1,2,1,5,2,2,
%T A290090 2,3,1,3,2,4,1,5,1,5,1,2,1,5,2,3,1,5,1,2,2,7,2,2,1,3,1,3,3,6,2,4,1,3,
%U A290090 1,5,1,4,1,3,2,5,3,4,1,5,1,3,1,8,1,2,1,7,1,2,3,3,2,3,2,6,1,5,2,5,1,2,1,7,4,2,1,3,1,5,2,9,1,4,1,3,2,3,2,4
%N A290090 a(n) is the number of proper divisors of n that are odious (A000069).
%C A290090 If n is odd and k >= 1, then a(2^k*n) = (k+1)*n+k if n is in A000069 and (k+1)*n if n is not in A000069. - _Robert Israel_, Oct 03 2017
%H A290090 Antti Karttunen, <a href="/A290090/b290090.txt">Table of n, a(n) for n = 1..16385</a>
%H A290090 <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>
%F A290090 a(n) = Sum_{d|n, d<n} A010060(d).
%F A290090 a(n) = A227872(n) - A010060(n).
%F A290090 a(n) = A007814(A293231(n)).
%F A290090 A000035(a(n)) = A000035(A292257(n)). [Parity-wise equivalent with A292257.]
%e A290090 For n = 55 whose proper divisors are 1, 5 and 11 (in binary "1", "101" and "1011"), only 1 and 11 have an odd number of 1's in their binary representations, thus a(55) = 2.
%p A290090 f:= proc(n) nops(select(t -> convert(convert(t,base,2),`+`)::odd, numtheory:-divisors(n) minus {n})) end proc:
%p A290090 map(f, [$1..200]); # _Robert Israel_, Oct 03 2017
%t A290090 Table[DivisorSum[n, 1 &, And[OddQ@ DigitCount[#, 2, 1], # < n] &], {n, 120}] (* _Michael De Vlieger_, Oct 03 2017 *)
%o A290090 (PARI) A290090(n) = sumdiv(n,d,(d<n)*(hammingweight(d)%2));
%Y A290090 Cf. A000005, A000069, A010060, A227872, A292257, A293231.
%K A290090 nonn,base
%O A290090 1,4
%A A290090 _Antti Karttunen_, Oct 03 2017