login
A093688
Numbers m such that all divisors of m, excluding the divisor 1, have an even number of 1's in their binary expansions.
10
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
KEYWORD
nonn,base,easy
AUTHOR
Jason Earls, May 16 2004
EXTENSIONS
a(1) added by Charles R Greathouse IV, Mar 28 2013
STATUS
approved