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A353236
Number of divisors of n whose arithmetic derivative is even.
3
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
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
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