A227872 Number of odious divisors (A000069) of n.

1, 2, 1, 3, 1, 2, 2, 4, 1, 2, 2, 3, 2, 4, 1, 5, 1, 2, 2, 3, 3, 4, 1, 4, 2, 4, 1, 6, 1, 2, 2, 6, 2, 2, 3, 3, 2, 4, 2, 4, 2, 6, 1, 6, 1, 2, 2, 5, 3, 4, 1, 6, 1, 2, 3, 8, 2, 2, 2, 3, 2, 4, 3, 7, 2, 4, 2, 3, 2, 6, 1, 4, 2, 4, 2, 6, 3, 4, 2, 5, 2, 4, 1, 9, 1, 2, 2

Offset: 1

Author

Vladimir Shevelev, Oct 25 2013

Keywords

• nonn
• base

Links

• Peter J. C. Moses, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000069, A093688, A093696, A129771, A330289, A355968, A355969, A227873 (sum odious divs.).

Cf. A010060.

Programs

Maple

A227872 := proc(n)
    option remember ;
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if not isA001969(d) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A227872(n),n=1..200) ; # R. J. Mathar, Aug 07 2022

Mathematica

a[n_] := DivisorSum[n, 1 &, OddQ[DigitCount[#, 2, 1]] &]; Array[a, 100] (* Amiram Eldar, Jul 23 2022 *)

PARI

a(n) = sumdiv(n, d, hammingweight(d) % 2); \\ Michel Marcus, Feb 06 2016

PARI

isod(n) = hammingweight(n) % 2; \\ A000069
a(n) = my(v=valuation(n, 2)); n >>= v; sumdiv(n,d,isod(d)) * (v+1) \\ David A. Corneth, Jul 23 2022

Python

from sympy import divisors
def c(n): return bin(n).count("1")&1
def a(n): return sum(1 for d in divisors(n, generator=True) if c(d))
print([a(n) for n in range(1, 101)]) # Michael S. Branicky, Jul 23 2022

Formula

a(n) + A356018(n) = A000005(n).

a(2^n) = n+1. - Bernard Schott, Jul 22 2022

a(n) = 1 iff n is in A093688. - Bernard Schott, Jul 23 2022

a(n) = Sum_{d|n} A010060(d). - Ridouane Oudra, Apr 12 2025

Extensions

More terms from Peter J. C. Moses, Oct 25 2013