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

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, 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, 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

Offset: 1

Author

Antti Karttunen, Oct 03 2017

Keywords

• nonn
• base

Comments

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

Examples

			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.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16385
• Index entries for sequences related to binary expansion of n

Crossrefs

Cf. A000005, A000069, A010060, A227872, A292257, A293231.

Programs

Maple

f:= proc(n) nops(select(t -> convert(convert(t,base,2),`+`)::odd, numtheory:-divisors(n) minus {n})) end proc:
map(f, [$1..200]); # Robert Israel, Oct 03 2017

Mathematica

Table[DivisorSum[n, 1 &, And[OddQ@ DigitCount[#, 2, 1], # < n] &], {n, 120}] (* Michael De Vlieger, Oct 03 2017 *)

PARI

A290090(n) = sumdiv(n,d,(d
				

Formula

a(n) = Sum_{d|n, dA010060(d).

a(n) = A227872(n) - A010060(n).

a(n) = A007814(A293231(n)).

A000035(a(n)) = A000035(A292257(n)). [Parity-wise equivalent with A292257.]