A093688 Numbers m such that all divisors of m, excluding the divisor 1, have an even number of 1's in their binary expansions.

1, 3, 5, 9, 15, 17, 23, 27, 29, 43, 45, 51, 53, 71, 83, 85, 89, 101, 113, 129, 135, 139, 149, 153, 159, 163, 197, 215, 249, 255, 257, 263, 267, 269, 277, 281, 293, 303, 311, 317, 337, 347, 349, 353, 359, 373, 383, 387, 389, 401, 417, 447, 449, 459, 461, 467, 479

Offset: 1

Author

Jason Earls, May 16 2004

Keywords

• nonn
• base
• easy

Comments

Putting the 1 aside, these could be called odiousfree numbers, and are a subsequence of A001969. A093696 would be the evilfree numbers then. - Irina Gerasimova, Feb 08 2014.

Equivalently, numbers all of whose divisors > 1 are evil (A001969). - Bernard Schott, Jul 24 2022

Examples

			51 is in the sequence because, excluding 1, its divisors are 3, 17 and 51.
In binary: 11, 10001, 110011 all have an even number of 1's.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A001969, A093696, A356018, A356019, A356020.

Programs

Mathematica

okQ[n_] := AllTrue[Rest[Divisors[n]], EvenQ[Total[IntegerDigits[#, 2]]]&]; Select[Range[500], okQ] (* Jean-François Alcover, Dec 06 2015 *)

PARI

is(n)=sumdiv(n,d,hammingweight(d)%2)==1 \\ Charles R Greathouse IV, Mar 28 2013

Python

from sympy import divisors
def c(n): return n == 1 or bin(n).count("1")&1 == 0
def ok(n): return n > 0 and all(c(d) for d in divisors(n, generator=True))
print([k for k in range(480) if ok(k)]) # Michael S. Branicky, Jul 24 2022

Extensions

a(1) added by Charles R Greathouse IV, Mar 28 2013