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

1,2

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

Links

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

Example

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.

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A116649 A129771 A209837 * A143512 A174688 A339345

Adjacent sequences: A093685 A093686 A093687 * A093689 A093690 A093691

Keyword

nonn,base,easy

Author

Jason Earls, May 16 2004

Extensions

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

Status

approved