A227873 -id:A227873 - cp's OEIS frontend

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

Vladimir Shevelev, Oct 25 2013

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

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

Cf. A010060.

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

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

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

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

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

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

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

A227891 Numbers for which the number of odious proper divisors (A000069) equals the number of evil proper divisors (A001969).

Original entry on oeis.org

1, 9, 25, 289, 441, 529, 625, 841, 1849, 2809, 3249, 5041, 6889, 7225, 7569, 7921, 10201, 12769, 15129, 15625, 19321, 21025, 22201, 26569, 31329, 38809, 46225, 48841, 53361, 55225, 66049, 69169, 72361, 76729, 78961, 83521, 85849, 93025, 96721, 100489, 103041

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Oct 26 2013

All terms are odd squares (see Shevelev links).

			1 has no proper divisors, so it is in the sequence.
9 has two proper divisors 1 (odious) and 3 (evil). Thus 9 is in the sequence.

Donovan Johnson, Table of n, a(n) for n = 1..1000
Vladimir Shevelev, A set of sequences of perfect squares.
Vladimir Shevelev, A question of Donovan Johnson.

Cf. A000005, A227872, A227873, A227889, A000069, A001969.

isQ[n_] := Sum[Switch[Mod[Total[IntegerDigits[d, 2]], 2], 0, 1, 1, -1], {d, Most[Divisors[n]]}] == 0; Select[(2*Range[200]-1)^2, isQ] (* Jean-François Alcover, Dec 04 2015 *)

is(n)=sumdiv(n,d,(-1)^hammingweight(d))==(-1)^hammingweight(n)
select(is, vector(10^4,i,(2*i-1)^2)) \\ Charles R Greathouse IV, Oct 26 2013

c=0; forstep(i=1, 8135, 2, n=i^2; nd=numdiv(n); d=divisors(n); ce=0; co=0; for(j=1, nd-1, if(hammingweight(d[j])%2==0, ce++, co++)); if(ce==co, c++; write("b227891.txt", c " " n))) \\ Donovan Johnson, Oct 30 2013

Common value for numbers of considered divisors is (A000005(a(n))-1)/2.

A260934 Sum of evil divisors of n. For evil numbers see A001969.

Original entry on oeis.org

0, 0, 3, 0, 5, 9, 0, 0, 12, 15, 0, 21, 0, 0, 23, 0, 17, 36, 0, 35, 3, 0, 23, 45, 5, 0, 39, 0, 29, 69, 0, 0, 36, 51, 5, 84, 0, 0, 42, 75, 0, 9, 43, 0, 77, 69, 0, 93, 0, 15, 71, 0, 53, 117, 5, 0, 60, 87, 0, 161, 0, 0, 75, 0, 70

Offset: 1

Juri-Stepan Gerasimov, Aug 04 2015

a(n) = 0 if there is no evil divisor.

			a(6) = A000203(6) - A227873(6) = 12 - 3 = 9.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001969, A227873, A000203.

Total[Select[Divisors@ #, EvenQ@ First@ DigitCount[#, 2] &]] & /@ Range@ 65 (* Michael De Vlieger, Aug 04 2015 *)

a(n) = A000203(n) - A227873(n). - Vladimir Shevelev, Oct 25 2013.

Edited by Wolfdieter Lang, Aug 23 2015

A227889 Numbers for which sum of odious proper divisors (A000069) equals sum of evil proper divisors (A001969).

Original entry on oeis.org

6, 11346, 1721418, 7449858, 11215266, 14101830, 28118346, 31755786, 37118418, 48517386, 69016314, 78075906, 258216018, 409092018, 410775306, 443414418, 453980706, 471867666, 525843960, 582427266, 758573106, 800349666, 805060626, 874923018, 1042069218, 1458081714

Offset: 1

Vladimir Shevelev and Peter J. C. Moses, Oct 26 2013

Are there terms not divisible by 6?

All the displayed terms are an odd multiple of 6, and up to a few exceptions of the form a(n)=6*p*q, where p,q have the same odd Hamming weight H(p) = H(q) >= 7. - M. F. Hasler, Oct 27 2013

			6 has odious divisors 1,2 and proper evil divisor 3. Since 1+2=3, then 6 is in the sequence.

Cf. A000203, A227872, A227873, A000069, A001969.

for(n=4, 1458081714, if(isprime(n), next); nd=numdiv(n); if(nd>3, d=divisors(n); se=0; so=1; for(j=2, nd-1, if(hammingweight(d[j])%2==0, se=se+d[j], so=so+d[j])); if(se==so, print1(n ", ")))) /* Donovan Johnson, Oct 26 2013 */

is(n,d=divisors(n))={sum(j=2, #d-1, (-1)^hammingweight(d[j])*d[j])==1} \\ - M. F. Hasler, Oct 27 2013

Common value of the considered sums of divisors is (A000203(a(n))-a(n))/2.

a(5)-a(26) from Donovan Johnson, Oct 26 2013

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A227872 Number of odious divisors (A000069) of n.

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A227891 Numbers for which the number of odious proper divisors (A000069) equals the number of evil proper divisors (A001969).

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A260934 Sum of evil divisors of n. For evil numbers see A001969.

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A227889 Numbers for which sum of odious proper divisors (A000069) equals sum of evil proper divisors (A001969).

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