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

1,4

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

Links

Antti Karttunen, Table of n, a(n) for n = 1..16385

Index entries for sequences related to binary expansion of n

Formula

a(n) = Sum_{d|n, d<n} A010060(d).

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

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

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

Example

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.

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 *)

Prog

(PARI) A290090(n) = sumdiv(n, d, (d<n)*(hammingweight(d)%2));

Crossrefs

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

Sequence in context: A305079 A319841 A336099 * A066075 A359211 A072347

Adjacent sequences: A290087 A290088 A290089 * A290091 A290092 A290093

Keyword

nonn,base

Author

Antti Karttunen, Oct 03 2017

Status

approved