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

1,2

Comments

Subsequence of A000069. - Michel Marcus, Feb 09 2014

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

Links

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

Formula

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

Example

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.

Maple

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

Mathematica

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

Prog

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

Crossrefs

Cf. A000069, A001969, A227872, A330289, A355968, A355969.

Similar sequences: A062687, A190217, A337741, A337941, A355596.

A000079 is a subsequence.

Sequence in context: A089559 A165741 A238989 * A190851 A325427 A336656

Adjacent sequences: A093693 A093694 A093695 * A093697 A093698 A093699

Keyword

nonn,base

Author

Jason Earls, May 16 2004

Status

approved