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Number of divisors of n whose arithmetic derivative is even.
3

%I #19 May 13 2024 09:14:51

%S 1,1,1,2,1,1,1,3,2,1,1,3,1,1,2,4,1,2,1,3,2,1,1,5,2,1,2,3,1,2,1,5,2,1,

%T 2,5,1,1,2,5,1,2,1,3,3,1,1,7,2,2,2,3,1,2,2,5,2,1,1,6,1,1,3,6,2,2,1,3,

%U 2,2,1,8,1,1,3,3,2,2,1,7,3,1,1,6,2,1,2,5,1,3,2,3,2,1,2,9,1,2,3,5,1,2,1,5,4

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

%C Number of terms of A235992 that divide n. - _Antti Karttunen_, May 13 2024

%H Antti Karttunen, <a href="/A353236/b353236.txt">Table of n, a(n) for n = 1..65537</a>

%F a(n) = Sum_{d|n} ((1+d') mod 2).

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

%F a(n) = A000005(n) - A353235(n).

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

%e a(12) = 3; 12 has 3 divisors whose arithmetic derivatives are even: 1' = 0, 4' = 4, and 12' = 16.

%t d[1] = 0; d[n_] := n*Plus @@ ((Last[#]/First[#]) & /@ FactorInteger[n]); a[n_] := DivisorSum[n, 1 &, EvenQ[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'), A235992, A353235, A353237.

%Y Inverse Möbius transform of A358680.

%K nonn

%O 1,4

%A _Wesley Ivan Hurt_, May 01 2022

%E Data section extended up to a(105) by _Antti Karttunen_, May 13 2024