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

%I A093696 #31 Aug 07 2022 08:38:34
%S A093696 1,2,4,7,8,11,13,14,16,19,22,26,28,31,32,37,38,41,44,47,49,52,56,59,
%T A093696 61,62,64,67,73,74,76,79,82,88,91,94,97,98,103,104,107,109,112,118,
%U A093696 121,122,124,127,128,131,133,134,137,143,146,148,151,152,157,158,164,167,173
%N A093696 Numbers n such that all divisors of n have an odd number of 1's in their binary expansions.
%C A093696 Subsequence of A000069. - _Michel Marcus_, Feb 09 2014
%C A093696 Numbers all of whose divisors are odious. - _Bernard Schott_, Jul 22 2022
%H A093696 Amiram Eldar, <a href="/A093696/b093696.txt">Table of n, a(n) for n = 1..10000</a>
%F A093696 {n: A356018(n) =0 }. - _R. J. Mathar_, Aug 07 2022
%e A093696 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.
%p A093696 isA001969 := proc(n)
%p A093696     if wt(n) mod 2 = 0 then
%p A093696         true;
%p A093696     else
%p A093696         false;
%p A093696     end if;
%p A093696 end proc:
%p A093696 isA093696 := proc(n)
%p A093696     for d in numtheory[divisors](n) do
%p A093696         if isA001969(d) then
%p A093696             return false;
%p A093696         end if;
%p A093696     end do;
%p A093696     true;
%p A093696 end proc:
%p A093696 for n from 1 to 200 do
%p A093696     if isA093696(n) then
%p A093696         printf("%d,",n);
%p A093696     end if;
%p A093696 end do: # _R. J. Mathar_, Feb 13 2014
%t A093696 odiousQ[n_] := OddQ @ DigitCount[n, 2][[1]]; Select[Range[200], AllTrue[ Divisors[#], odiousQ ] &] (* _Amiram Eldar_, Dec 09 2019 *)
%o A093696 (PARI) is(n)=fordiv(n,d,if(hammingweight(d)%2==0, return(0))); 1 \\ _Charles R Greathouse IV_, Mar 29 2013
%o A093696 (Python)
%o A093696 from sympy import divisors, isprime
%o A093696 def c(n): return bin(n).count("1")&1
%o A093696 def ok(n): return n > 0 and all(c(d) for d in divisors(n, generator=True))
%o A093696 print([k for k in range(174) if ok(k)]) # _Michael S. Branicky_, Jul 24 2022
%Y A093696 Cf. A000069, A001969, A227872, A330289, A355968, A355969.
%Y A093696 Similar sequences: A062687, A190217, A337741, A337941, A355596.
%Y A093696 A000079 is a subsequence.
%K A093696 nonn,base
%O A093696 1,2
%A A093696 _Jason Earls_, May 16 2004