A355596 -id:A355596 - cp's OEIS frontend

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

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

A355593 a(n) is the number of alternating integers that divide n.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 1, 6, 1, 4, 3, 5, 1, 6, 1, 5, 4, 2, 2, 7, 3, 2, 4, 5, 2, 7, 1, 6, 2, 3, 3, 9, 1, 3, 2, 6, 2, 7, 2, 3, 5, 3, 2, 8, 3, 6, 2, 4, 1, 8, 2, 7, 2, 4, 1, 9, 2, 2, 6, 6, 3, 4, 2, 4, 4, 7, 1, 11, 1, 3, 4, 5, 2, 5, 1, 7, 5, 3, 2, 9, 3, 3, 4, 4, 2, 11, 2, 5, 2, 4, 2, 10, 1, 6, 3, 7

Offset: 1

Bernard Schott, Jul 08 2022

This sequence first differs from A355302 at index 13, where a(13) = 1 while A355302(13) = 2.

This sequence first differs from A332268 at index 14, where a(14) = 4 while A332268(14) = 3.

			40 has 8 divisors: {1, 2, 4, 5, 8, 10, 20, 40} of which 2 are not alternating integers: {20, 40}, hence a(40) = 8 - 2 = 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A030141 (alternating integers), A355594, A355595, A355596.

Similar to A332268 (with Niven numbers) and A355302 (with undulating integers).

Alt:= [$1..9, seq(seq(10*i+r - (i mod 2), r=[1,3,5,7,9]),i=1..9)]:
V:= Vector(100):
for t in Alt do J:= [seq(i,i=t..100,t)]; V[J]:= V[J] +~ 1 od:
convert(V,list); # Robert Israel, Nov 26 2023

q[n_] := !MemberQ[Differences[Mod[IntegerDigits[n], 2]], 0]; a[n_] := DivisorSum[n, 1 &, q[#] &]; Array[a, 120] (* Amiram Eldar, Jul 08 2022 *)

alternate(n,d=digits(n))=for(i=2,#d, if((d[i]-d[i-1])%2==0, return(0))); 1
a(n)=sumdiv(n,d,alternate(d)) \\ Charles R Greathouse IV, Jul 08 2022

from sympy import divisors
def p(d): return 0 if d in "02468" else 1
def c(n):
    if n < 10: return True
    s = str(n)
    return all(p(s[i]) != p(s[i+1]) for i in range(len(s)-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 08 2022

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=2} 1/A030141(n) = 5.1... (the sums up to 10^10, 10^11 and 10^12 are 5.1704..., 5.1727... and 5.1738..., respectively). - Amiram Eldar, Jan 06 2024

A355594 a(n) is the smallest integer that has exactly n alternating divisors.

Original entry on oeis.org

1, 2, 4, 6, 16, 12, 24, 48, 36, 96, 72, 144, 210, 180, 420, 360, 504, 864, 630, 1080, 1512, 2160, 1260, 3150, 1890, 2520, 5040, 6300, 3780, 10080, 12600, 9450, 7560, 32760, 15120, 18900, 22680, 30240, 88830, 37800, 45360, 75600, 105840, 90720, 151200, 162540, 254520

Offset: 1

Bernard Schott, Jul 08 2022

This sequence first differs from A005179 at index 7 where A005179(7) = 64.

			16 has 5 divisors: {1, 2, 4, 8, 16} all of which are alternating integers; no positive integer smaller than 16 has five alternating divisors, hence a(5) = 16.
96 has 12 divisors: {1, 2, 3, 4, 6, 8, 12, 16, 24, 32, 48, 96}, only 24 and 48 are not alternating; no positive integer smaller than 96 has ten alternating divisors, hence a(10) = 96.

David A. Corneth, Table of n, a(n) for n = 1..147 (first 107 terms from Robert Israel)
David A. Corneth, Some upper bounds on a(n)

Cf. A005179, A030141 (alternating numbers), A355593, A355595, A355596.

Similar, but with undulating divisors: A355303.

isalt:= proc(n) local L; option remember;
   L:= convert(n,base,10) mod 2;
   L:= L[2..-1]-L[1..-2];
   not member(0,L)
end proc:
N:= 50: # for a(1)..a(N)
V:= Vector(N): count:= 0:
for n from 1 while count < N do
  w:= nops(select(isalt,numtheory:-divisors(n)));
  if w <= N and V[w] = 0 then V[w]:= n; count:= count+1 fi
od:
convert(V,list); # Robert Israel, Jan 24 2023

q[n_] := ! MemberQ[Differences[Mod[IntegerDigits[n], 2]], 0]; f[n_] := DivisorSum[n, 1 &, q[#] &]; 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[50, 10^6] (* Amiram Eldar, Jul 08 2022 *)

is(n, d=digits(n))=for(i=2, #d, if((d[i]-d[i-1])%2==0, return(0))); 1; \\ A030141
a(n) = my(k=1); while (sumdiv(k, d, is(d)) != n, k++); k; \\ Michel Marcus, Jul 11 2022

from itertools import count
from sympy import divisors
def A355594(n):
    for m in count(1):
        if sum(1 for k in divisors(m,generator=True) if all(int(a)+int(b)&1 for a, b in zip(str(k),str(k)[1:]))) == n:
            return m # Chai Wah Wu, Jul 12 2022

a(n) >= A005179(n). - David A. Corneth, Jan 25 2023

More terms from David A. Corneth, Jul 08 2022

cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A355593 a(n) is the number of alternating integers that divide n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A355594 a(n) is the smallest integer that has exactly n alternating divisors.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

Extensions