A353236 Number of divisors of n whose arithmetic derivative is even.

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, 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, 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

Offset

1,4

Comments

Number of terms of A235992 that divide n. - Antti Karttunen, May 13 2024

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

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

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

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

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

Example

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

Mathematica

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 *)

Prog

(PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = sumdiv(n, d, !(ad(d) % 2)); \\ Michel Marcus, May 02 2022

Crossrefs

Cf. A000005 (tau), A003415 (n'), A235992, A353235, A353237.

Inverse Möbius transform of A358680.

Sequence in context: A181591 A347442 A336424 * A325939 A318586 A222580

Adjacent sequences: A353233 A353234 A353235 * A353237 A353238 A353239

Keyword

nonn

Author

Wesley Ivan Hurt, May 01 2022

Extensions

Data section extended up to a(105) by Antti Karttunen, May 13 2024

Status

approved