A227872 -id:A227872 - cp's OEIS frontend

A355969 Positions of records in A227872, i.e., integers whose number of odious divisors sets a new record.

1, 2, 4, 8, 16, 28, 56, 84, 112, 168, 336, 672, 1344, 2184, 4368, 8736, 17472, 30576, 34944, 41664, 48048, 61152, 80080, 83328, 96096, 122304, 160160, 192192, 240240, 320320, 336336, 480480, 672672, 960960, 1345344, 1681680, 1921920, 2489760, 2690688, 2738736

Offset: 1

Bernard Schott, Jul 22 2022

Corresponding records of number of odious divisors are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, ...

			a(7) = 56 is in the sequence because A227872(56) = 8 is larger than any earlier value in A227872.

Cf. A000069, A093696, A227872, A330289, A355968.

Similar sequences: A093036, A093037, A330815, A340549.

f[n_] := DivisorSum[n, 1 &, OddQ[DigitCount[#, 2, 1]] &]; fm = -1; s = {}; Do[If[(fn = f[n]) > fm, fm = fn; AppendTo[s, n]], {n, 1, 10^5}]; s (* Amiram Eldar, Jul 22 2022 *)

lista(nn)= my(list = List(), m=0, new); for (n=1, nn, new = sumdiv(n, d, isod(d)); if (new > m, listput(list, n); m = new);); Vec(list); \\ Michel Marcus, Jul 22 2022

from sympy import divisors
from itertools import count, islice
def c(n): return bin(n).count("1")&1
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen(record=-1):
    for k in count(1):
        if f(k) > record: record = f(k); yield k
print(list(islice(agen(), 30))) # Michael S. Branicky, Jul 23 2022

More terms from Amiram Eldar, Jul 22 2022

A357761 a(n) = A227872(n) - A356018(n).

Original entry on oeis.org

1, 2, 0, 3, 0, 0, 2, 4, -1, 0, 2, 0, 2, 4, -2, 5, 0, -2, 2, 0, 2, 4, 0, 0, 1, 4, -2, 6, 0, -4, 2, 6, 0, 0, 2, -3, 2, 4, 0, 0, 2, 4, 0, 6, -4, 0, 2, 0, 3, 2, -2, 6, 0, -4, 2, 8, 0, 0, 2, -6, 2, 4, 0, 7, 0, 0, 2, 0, 0, 4, 0, -4, 2, 4, -2, 6, 2, 0, 2, 0, -1, 4, 0

Offset: 1

Amiram Eldar, Oct 12 2022

The excess of the number of odious (A000069) divisors of n over the number of evil (A001969) divisors of n.

Every integer occurs in this sequence.

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

Cf. A000005, A000069, A000290 (positions of odd terms), A001969, A027697, A027699, A106400, A227872, A230851 (positions of 0's), A356018, A357762.

Similar sequences: A046660, A048272.

a[n_] := -DivisorSum[n, (-1)^DigitCount[#, 2, 1] &]; Array[a, 100]

a(n) = -sumdiv(n, d, (-1)^hammingweight(d));

a(n) = -Sum_{d|n} A106400(d).
a(n) = A000005(n) - 2*A356018(n).
a(n) = 2*A227872(n) - A000005(n).
a(n) = 0 iff n is in A230851.
a(n) == 1 (mod 2) iff n is a square (A000290).
a(2^n) = n + 1.
a(p*2^n) = 0 when p is an evil prime (A027699).
a(p^2*2^n) = n + 1 when p is an evil prime (A027699) and p^2 is odious, and when p is an odd odious prime (A027697) and p^2 is evil.
a(p^2*2^n) = -(n+1) when p is an evil prime and p^2 is also evil.
a(p^2*2^n) = 3*(n+1) when p is an odd odious prime and p^2 is also odious.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = -Sum_{k>=1} A106400(k)/k = 1.196283264... (A357762).

A356040 a(n) is the smallest integer that has exactly n odious divisors (A227872) and n evil divisors (A356018).

Original entry on oeis.org

3, 6, 12, 24, 48, 96, 192, 210, 252, 528, 3072, 420, 12288, 2112, 1008, 840, 196608, 2016, 786432, 1680, 4032, 33792, 12582912, 3360, 30000, 135168, 16128, 6720, 805306368, 19152, 3221225472, 13440, 64512, 2162688, 120000, 26880, 206158430208, 8650752, 258048, 31920

Offset: 1

Bernard Schott, Jul 24 2022

As the number of divisors is even, there is no square in the sequence.

			48 has ten divisors, five of which are odious {1, 2, 4, 8, 16} as they have an odd number of 1's in their binary expansion: 1, 10, 100, 1000 and 10000; the five other divisors are evil {3, 6, 12, 24, 48} as they have an even number of 1's in their binary expansion: 11, 110, 1100, 11000 and 110000; also, no positive integer smaller than 48 has five divisors that are evil and five divisors that are odious, hence a(5) = 48.

David A. Corneth, Table of n, a(n) for n = 1..3322

Cf. A000069, A001969, A227872, A356018.

Cf. A230851, A005179.

f[n_] := DivisorSum[n, {1, (-1)^DigitCount[#, 2][[1]]} &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i, d}, While[c < len && n < nmax, i = f[n]; If[i[[2]] == 0, d = i[[1]]/2; If[d <= len && s[[d]] == 0, c++; s[[d]] = n]]; n++]; s]; seq[16, 10^6] (* Amiram Eldar, Jul 24 2022 *)

a(n) = if(isprime(n), return(2^(n-1)*3)); forfactored(i=1, 2^(n-1)*3, if(numdiv(i[2]) == 2*n, d=divisors(i[2]); if(sum(j=1, #d, isevil(d[j])) == n, return(i[1]))))
isevil(n) = bitand(hammingweight(n), 1) == 0 \\ David A. Corneth, Jul 24 2022

from sympy import divisors, isprime
from itertools import count, islice
def f(n):
    counts = [0, 0]
    for d in divisors(n, generator=True):
        counts[bin(d).count("1")&1] += 1
    return counts[0] if counts[0] == counts[1] else -1
def a(n):
    if isprime(n): return 2**(n-1) * 3
    return next(k for k in count(1) if f(k) == n)
print([a(n) for n in range(1, 34)]) # Michael S. Branicky, Jul 24 2022

a(n) <= 2^(n-1) * 3.
a(n) = 2^(n-1) * 3 if n is prime. - David A. Corneth, Jul 24 2022
a(n) >= A005179(2*n). - Michael S. Branicky, Jul 24 2022

a(9)-a(37) from Amiram Eldar, Jul 24 2022

a(38)-a(40) from David A. Corneth, Jul 24 2022

A093696 Numbers n such that all divisors of n have an odd number of 1's in their binary expansions.

Original entry on oeis.org

1, 2, 4, 7, 8, 11, 13, 14, 16, 19, 22, 26, 28, 31, 32, 37, 38, 41, 44, 47, 49, 52, 56, 59, 61, 62, 64, 67, 73, 74, 76, 79, 82, 88, 91, 94, 97, 98, 103, 104, 107, 109, 112, 118, 121, 122, 124, 127, 128, 131, 133, 134, 137, 143, 146, 148, 151, 152, 157, 158, 164, 167, 173

Offset: 1

Jason Earls, May 16 2004

Subsequence of A000069. - Michel Marcus, Feb 09 2014

Numbers all of whose divisors are odious. - Bernard Schott, Jul 22 2022

			14 is in the sequence because its divisors are [1, 2, 7, 14] and in binary: 1, 10, 111 and 1110, all have an odd number of 1's.

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

Cf. A000069, A001969, A227872, A330289, A355968, A355969.
Similar sequences: A062687, A190217, A337741, A337941, A355596.
A000079 is a subsequence.

isA001969 := proc(n)
    if wt(n) mod 2 = 0 then
        true;
    else
        false;
    end if;
end proc:
isA093696 := proc(n)
    for d in numtheory[divisors](n) do
        if isA001969(d) then
            return false;
        end if;
    end do;
    true;
end proc:
for n from 1 to 200 do
    if isA093696(n) then
        printf("%d,",n);
    end if;
end do: # R. J. Mathar, Feb 13 2014

odiousQ[n_] := OddQ @ DigitCount[n, 2][[1]]; Select[Range[200], AllTrue[ Divisors[#], odiousQ ] &] (* Amiram Eldar, Dec 09 2019 *)

is(n)=fordiv(n,d,if(hammingweight(d)%2==0, return(0))); 1 \\ Charles R Greathouse IV, Mar 29 2013

from sympy import divisors, isprime
def c(n): return bin(n).count("1")&1
def ok(n): return n > 0 and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(174) if ok(k)]) # Michael S. Branicky, Jul 24 2022

{n: A356018(n) =0 }. - R. J. Mathar, Aug 07 2022

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.

A356018 a(n) is the number of evil divisors (A001969) of n.

Original entry on oeis.org

0, 0, 1, 0, 1, 2, 0, 0, 2, 2, 0, 3, 0, 0, 3, 0, 1, 4, 0, 3, 1, 0, 1, 4, 1, 0, 3, 0, 1, 6, 0, 0, 2, 2, 1, 6, 0, 0, 2, 4, 0, 2, 1, 0, 5, 2, 0, 5, 0, 2, 3, 0, 1, 6, 1, 0, 2, 2, 0, 9, 0, 0, 3, 0, 2, 4, 0, 3, 2, 2, 1, 8, 0, 0, 4, 0, 1, 4, 0, 5, 3, 0, 1, 3, 3, 2

Offset: 1

Bernard Schott, Jul 23 2022

a(n) = 0 iff n is in A093696.

			12 has 6 divisors: {1, 2, 3, 4, 6, 12} of which three {3, 6, 12} have an even number of 1's in their binary expansion with 11, 110 and 11100 respectively; hence a(12) = 3.

Cf. A000005, A001969, A093688, A093696 (location of 0s), A227872, A356019, A356020.

Similar sequences: A083230, A087990, A087991, A332268, A355302.

A356018 := proc(n)
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if isA001969(d) then
            a := a+1 ;
        end if;
    end do:
    a ;
end proc:
seq(A356018(n),n=1..200) ;  # R. J. Mathar, Aug 07 2022

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

a(n) = my(v = valuation(n, 2)); n>>=v; d=divisors(n); sum(i=1, #d, bitand(hammingweight(d[i]), 1) == 0) * (v+1) \\ David A. Corneth, Jul 23 2022

from sympy import divisors
def c(n): return bin(n).count("1")&1 == 0
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) = A000005(n) - A227872(n).

More terms from David A. Corneth, Jul 23 2022

A227873 Sum of odious divisors of n. See A000069 for odious numbers.

Original entry on oeis.org

1, 3, 1, 7, 1, 3, 8, 15, 1, 3, 12, 7, 14, 24, 1, 31, 1, 3, 20, 7, 29, 36, 1, 15, 26, 42, 1, 56, 1, 3, 32, 63, 12, 3, 43, 7, 38, 60, 14, 15, 42, 87, 1, 84, 1, 3, 48, 31, 57, 78, 1, 98, 1, 3, 67, 120, 20, 3, 60, 7, 62, 96, 29, 127, 14, 36, 68, 7, 70, 129, 1, 15

Offset: 1

Vladimir Shevelev, Oct 25 2013

Sum of evil divisors of n is A000203(n) - a(n) = A260934(n). See A001969 for evil numbers.

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

Cf. A000203, A227872, A000069, A001969, A212302, A260934, A010060, A102392.

A227873 := proc(n)
    option remember ;
    local a,d ;
    a := 0 ;
    for d in numtheory[divisors](n) do
        if not isA001969(d) then
            a := a+d ;
        end if;
    end do:
    a ;
end proc:
seq(A227873(n),n=1..200) ; # R. J. Mathar, Aug 17 2022

Total[Select[Divisors@ #, OddQ@ First@ DigitCount[#, 2] &]] & /@ Range@ 72 (* Michael De Vlieger, Aug 04 2015 *)

a(n) = sumdiv(n, d, d*(hammingweight(d) % 2)); \\ Michel Marcus, Aug 04 2015

a(n) = Sum_{d|n} A102392(d). - Ridouane Oudra, Apr 19 2025

More terms from Peter J. C. Moses

Minor changes. - Wolfdieter Lang, Aug 23 2015

A355968 a(n) is the smallest number that has exactly n odious divisors (A000069).

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 64, 56, 84, 112, 1024, 168, 4096, 448, 336, 728, 36309, 672, 57057, 1456, 1344, 7168, 105105, 2184, 6384, 24150, 5376, 5208, 405405, 4368, 389025, 11648, 20020, 72618, 10416, 8736, 927675, 114114, 48300, 24024, 855855, 17472, 1426425, 40040

Offset: 1

Bernard Schott, Jul 21 2022

a(n) <= 2^(n-1) with equality for n = 1, 2, 3, 4, 5, 7, 11, 13 up to a(44).

			a(6) = 28 since 28 has 6 divisors {1, 2, 4, 7, 14, 28} that have all an odd number of 1's in their binary expansion: 1, 10, 100, 111, 1110 and 11100; also, no positive integer smaller than 28 has six divisors that are odious.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000069, A093696, A227872, A330289, A355969.

Similar sequences: A087997, A333456, A355303, A355699.

f[n_] := DivisorSum[n, 1 &, OddQ[DigitCount[#, 2, 1]] &]; seq[len_, nmax_] := Module[{s = Table[0, {len}], c = 0, n = 1, i}, While[c < len && n < nmax, i = f[n]; If[i <= len && s[[i]] == 0, c++; s[[i]] = n]; n++]; s]; seq[20, 10^6] (* Amiram Eldar, Jul 21 2022 *)

isod(n) = hammingweight(n) % 2; \\ A000069
a(n) = my(k=1); while (sumdiv(k, d, isod(d)) != n, k++); k; \\ Michel Marcus, Jul 22 2022

from sympy import divisors
from itertools import count, islice
def c(n): return bin(n).count("1")&1
def f(n): return sum(1 for d in divisors(n, generator=True) if c(d))
def agen():
    n, adict = 1, dict()
    for k in count(1):
        fk = f(k)
        if fk not in adict: adict[fk] = k
        while n in adict: yield adict[n]; n += 1
print(list(islice(agen(), 36))) # Michael S. Branicky, Jul 25 2022

More terms from Amiram Eldar, Jul 21 2022

A230851 Numbers with divisors which are half odious (A000069) and half evil (A001969).

Original entry on oeis.org

3, 5, 6, 10, 12, 17, 20, 23, 24, 29, 33, 34, 39, 40, 43, 46, 48, 53, 57, 58, 63, 65, 66, 68, 69, 71, 78, 80, 83, 86, 87, 89, 92, 95, 96, 101, 105, 106, 111, 113, 114, 115, 116, 117, 119, 123, 125, 126, 130, 132, 136, 138, 139, 141, 142, 145, 149, 156, 160, 163, 166, 171, 172, 174, 177, 178, 183

Offset: 1

Juri-Stepan Gerasimov, Oct 31 2013

nonn

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

Cf. A000069, A001969, A167171, A227872.

aQ[n_] := DivisorSum[n, (-1)^DigitCount[#, 2][[1]] &] == 0; Select[Range[200], aQ] (* Amiram Eldar, Sep 23 2019 *)

is(n)=!sumdiv(n,d,(-1)^hammingweight(d)) \\ Charles R Greathouse IV, Oct 31 2013

Numbers n such that d(n) = 2*A227872(n) where A227872(n) is number of odious divisors of n.

Corrected by Charles R Greathouse IV, Oct 31 2013

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

Original entry on oeis.org

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

Antti Karttunen, Oct 03 2017

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

			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.

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

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

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

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
				
```

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.]

cp's OEIS Frontend

A355969 Positions of records in A227872, i.e., integers whose number of odious divisors sets a new record.

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A357761 a(n) = A227872(n) - A356018(n).

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A356040 a(n) is the smallest integer that has exactly n odious divisors (A227872) and n evil divisors (A356018).

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A093696 Numbers n such that all divisors of n have an odd number of 1's in their binary expansions.

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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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A356018 a(n) is the number of evil divisors (A001969) of n.

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A227873 Sum of odious divisors of n. See A000069 for odious numbers.