A292257 - OEIS

%I #16 Oct 11 2024 19:59:07

%S 0,1,1,2,1,4,1,3,3,4,1,7,1,5,5,4,1,8,1,7,6,5,1,10,3,5,5,9,1,14,1,5,6,

%T 4,6,13,1,5,6,10,1,15,1,9,11,6,1,13,4,9,5,9,1,14,6,13,6,6,1,23,1,7,11,

%U 6,6,14,1,7,7,15,1,18,1,5,12,9,7,16,1,13,9,5,1,24,5,6,7,13,1,26,7,11,8,7,6,16,1,11,10,15,1,14,1,13,18

%N a(n) is the total number of 1's in binary expansion of all proper divisors of n.

%C If a(n) == A000120(n), then n is in A175522, if a(n) < A000120(n), then n is in A175524, and if a(n) > A000120(n), then n is in A175526.

%H Antti Karttunen, <a href="/A292257/b292257.txt">Table of n, a(n) for n = 1..16384</a>

%H <a href="/index/Bi#binary">Index entries for sequences related to binary expansion of n</a>.

%F a(n) = Sum_{d|n, d<n} A000120(d).

%F a(n) = A093653(n) - A000120(n).

%F a(n) = A192895(n) + A000120(n).

%F a(n) = A001222(A293214(n)).

%F A000035(a(n)) = A000035(A290090(n)). [Parity-wise equivalent with A290090.]

%t a[n_] := DivisorSum[n, DigitCount[#, 2, 1] &, # < n &]; Array[a, 100] (* _Amiram Eldar_, Jul 20 2023 *)

%t Table[Total[Flatten[IntegerDigits[#,2]&/@Most[Divisors[n]]]],{n,120}] (* _Harvey P. Dale_, Oct 11 2024 *)

%o (PARI) A292257(n) = sumdiv(n,d,(d<n)*hammingweight(d));

%Y Cf. A000120, A093653, A175522, A175524, A175526, A192895, A290090, A293214.

%K nonn,base

%O 1,4

%A _Antti Karttunen_, Oct 04 2017