A330289 -id:A330289 - 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

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

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

Original entry on oeis.org

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

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

A357763 Numbers m such that A357761(m) > A357761(k) for all k < m.

Original entry on oeis.org

1, 2, 4, 8, 16, 28, 56, 112, 224, 448, 728, 1456, 2912, 5824, 10192, 11648, 20384, 27664, 40768, 55328, 110656, 221312, 442624, 885248, 1263808, 1770496, 2527616, 3430336, 5055232, 6860672, 10110464, 13721344, 16155776, 20220928, 24012352, 32311552, 48024704

Offset: 1

Amiram Eldar, Oct 12 2022

First differs from A330289 at n = 28.
Since A357761(2^n) = n + 1, A357761 is unbounded and this sequence is infinite.
The corresponding record values are 1, 2, 3, 4, 5, 6, 8, 10, 12, 14, 16, 20, 24, 28, 30, 32, ... .

Amiram Eldar, Table of n, a(n) for n = 1..48

Cf. A330289, A355969, A356020, A357761, A357764.

f[n_] := -DivisorSum[n, (-1)^DigitCount[#, 2, 1] &]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 10^5}]; s

f(n) = -sumdiv(n, d, (-1)^hammingweight(d));
lista(nmax) = {my(fm = 0); for(n = 1, nmax, f1 = f(n); if(f1 > fm, fm = f1; print1(n, ", ")))};

A338611 Numbers all of whose divisors, excluding the divisor 1, are evil numbers (A001969) with a record number of divisors.

Original entry on oeis.org

1, 3, 9, 15, 45, 135, 765, 2295, 196605, 589815, 12884901885, 38654705655

Offset: 1

Amiram Eldar, Nov 03 2020

A number m is in this sequence if it is in A093688, and d(m) > d(k) for all terms k < m in A093688, where d(m) is the number of divisors of m (A000005).
The corresponding record numbers of divisors are 1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 64, ...
Apparently, all the terms except for 1 are products of powers of Fermat primes (A019434). 3 seems to be the only prime with multiplicity larger than 1 in some of the terms. There are no other terms in this sequence that are products of powers of the 5 known Fermat primes.

			The first 4 terms of A093688 are 1, 3, 5, 9, and their numbers of divisors are 1, 2, 2, 3. The record values 1, 2 and 3 occur at 1, 3 and 9 that are the first 3 terms of this sequence.

Subsequence of A001969 and A093688.
Similar sequence with odious numbers: A330289.
Cf. A000005, A019434.

evilQ[n_] := EvenQ @ DigitCount[n, 2, 1]; allDivEvilQ[n_] := AllTrue[Rest @ Divisors[n], evilQ]; divNumMax = 0; seq={}; Do[If[allDivEvilQ[n] && (divNum = DivisorSigma[0, n]) > divNumMax, divNumMax = divNum; AppendTo[seq, n]], {n, 1, 6*10^5}]; seq

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A227872 Number of odious divisors (A000069) of n.

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

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

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A357763 Numbers m such that A357761(m) > A357761(k) for all k < m.

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A338611 Numbers all of whose divisors, excluding the divisor 1, are evil numbers (A001969) with a record number of divisors.

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