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A246601 Sum of divisors d of n with property that the binary representation of d can be obtained from the binary representation of n by changing any number of 1's to 0's.

%I A246601 #33 Apr 09 2025 17:19:52
%S A246601 1,2,4,4,6,8,8,8,10,12,12,16,14,16,24,16,18,20,20,24,22,24,24,32,26,
%T A246601 28,40,32,30,48,32,32,34,36,36,40,38,40,43,48,42,44,44,48,60,48,48,64,
%U A246601 50,52,72,56,54,80,61,64,58,60,60,96,62,64,104,64,66,68,68,72,70,72
%N A246601 Sum of divisors d of n with property that the binary representation of d can be obtained from the binary representation of n by changing any number of 1's to 0's.
%C A246601 Equivalently, the sum of the divisors d of n such that the bitwise OR of d and n is equal to n. - _Chai Wah Wu_, Sep 06 2014
%C A246601 Equivalently, the sum of the divisors d of n such that the bitwise AND of n and d is equal to d. - _Amiram Eldar_, Dec 15 2022
%H A246601 N. J. A. Sloane, <a href="/A246601/b246601.txt">Table of n, a(n) for n = 1..10000</a>
%F A246601 a(2^i) = 2^i.
%F A246601 a(odd prime p) = p+1.
%F A246601 From _Amiram Eldar_, Dec 15 2022: (Start)
%F A246601 a(2*n) = 2*a(n), and therefore a(m*2^k) = 2^k*a(m) for m odd and k>=0.
%F A246601 a(2^n-1) = sigma(2^n-1) = A075708(n). (End)
%F A246601 a(n) = Sum_{d|n} d*(binomial(n,d) mod 2). - _Ridouane Oudra_, Apr 09 2025
%e A246601 12 = 1100_2; only the divisors 4 = 0100_2 and 12 = 1100_2 satisfy the condition, so(12) = 4+12 = 16.
%e A246601 15 = 1111_2; all divisors 1,3,5,15 satisfy the condition, so a(15)=24.
%p A246601 with(numtheory);
%p A246601 sd:=proc(n) local a,d,s,t,i,sw;
%p A246601 s:=convert(n,base,2);
%p A246601 a:=0;
%p A246601 for d in divisors(n) do
%p A246601 sw:=-1;
%p A246601 t:=convert(d,base,2);
%p A246601 for i from 1 to nops(t) do if t[i]>s[i] then sw:=1; fi; od:
%p A246601 if sw<0 then a:=a+d; fi;
%p A246601 od;
%p A246601 a;
%p A246601 end;
%p A246601 [seq(sd(n),n=1..100)];
%t A246601 a[n_] := DivisorSum[n, #*Boole[BitOr[#, n] == n] &]; Array[a, 100] (* _Jean-François Alcover_, Dec 02 2015, adapted from PARI *)
%o A246601 (Python)
%o A246601 from sympy import divisors
%o A246601 def A246601(n):
%o A246601     return sum(d for d in divisors(n) if n|d == n)
%o A246601 # _Chai Wah Wu_, Sep 06 2014
%o A246601 (PARI) a(n) = sumdiv(n, d, d*(bitor(n,d)==n)); \\ _Michel Marcus_, Sep 07 2014
%Y A246601 Cf. A000005, A000203, A075708, A246600.
%K A246601 nonn,base
%O A246601 1,2
%A A246601 _N. J. A. Sloane_, Sep 06 2014