%I #24 Jun 29 2024 22:52:10

%S 0,1,1,1,1,3,1,1,1,3,1,3,1,3,2,1,1,4,1,3,2,3,1,3,1,3,2,3,1,6,1,1,2,3,

%T 2,4,1,3,2,3,1,6,1,3,3,3,1,3,1,4,2,3,1,6,2,3,2,3,1,6,1,3,3,1,2,6,1,3,

%U 2,6,1,4,1,3,3,3,2,6,1,3,2,3,1,6,2,3,2,3,1,9,2,3,2,3

%N Number of divisors of n whose arithmetic derivative is odd.

%C Inverse Möbius transform of (n' mod 2), where n' is the arithmetic derivative of n (A003415). - _Wesley Ivan Hurt_, Jun 29 2024

%H Robert Israel, <a href="/A353235/b353235.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n} ((d') mod 2).

%F a(n) = tau(n)/2 - (1/2) * Sum_{d|n} (-1)^(d').

%F a(n) = A000005(n) - A353236(n).

%F a(n) = A000005(n)/2 - A353237(n)/2.

%F From _Robert Israel_, Jun 05 2023: (Start)

%F If n = 2^k * m where m is odd and k >= 1, a(n) = a(m) + A000005(m).

%F If n is odd and squarefree, a(n) = 2^(A001222(n)-1).

%F If p is an odd prime, a(p^k) = ceil(k/2).

%F If k and m are odd, a(k*m) = A000005(k)*a(m) + A000005(m)*a(k) - 2*a(m)*a(k).

%F (End)

%e a(12) = 3; 12 has 3 divisors whose arithmetic derivatives are odd: 2' = 1, 3' = 1, and 6' = 5.

%p aderodd:= proc(n) local t; option remember;

%p (n*add(t[2]/t[1],t=ifactors(n)[2]))::odd

%p end proc:

%p f:= proc(n) local t;

%p nops(select(aderodd, numtheory:-divisors(n)))

%p end proc:

%p map(f, [$1..100]); # _Robert Israel_, Jun 05 2023

%t d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, 1 &, OddQ[d[#]] &]; Array[a, 100] (* _Amiram Eldar_, May 02 2022 *)

%o (PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415

%o a(n) = sumdiv(n, d, ad(d) % 2); \\ _Michel Marcus_, May 02 2022

%Y Cf. A000005 (tau), A003415 (n'), A353236, A353237.

%K nonn

%O 1,6

%A _Wesley Ivan Hurt_, May 01 2022