A348281 Sum of the arithmetic derivatives of the squarefree divisors of n: a(n) = Sum_{d|n} d' * mu(d)^2.

0, 1, 1, 1, 1, 7, 1, 1, 1, 9, 1, 7, 1, 11, 10, 1, 1, 7, 1, 9, 12, 15, 1, 7, 1, 17, 1, 11, 1, 54, 1, 1, 16, 21, 14, 7, 1, 23, 18, 9, 1, 68, 1, 15, 10, 27, 1, 7, 1, 9, 22, 17, 1, 7, 18, 11, 24, 33, 1, 54, 1, 35, 12, 1, 20, 96, 1, 21, 28, 90, 1, 7, 1, 41, 10, 23, 20, 110, 1, 9, 1, 45

Offset

1,6

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000

Formula

a(p^k) = 1 for primes p and k >= 1. For k = 1, we have 1'*mu(1)^2 + p'*mu(p)^2 = 0*1 + 1*1 = 1. For all k >= 2, mu(p^k)^2 = 0. Therefore, a(p^k) = 0*1 + 1*1 + (0 + ... + 0) [k-1 times] = 1.

a(n) = Sum_{d|n} A008966(d) * A003415(d). - Antti Karttunen, Jan 13 2025

Example

a(10) = 9; a(10) = 1' + 2' + 5' + 10' = 0 + 1 + 1 + 7 = 9.

Prog

(PARI) ad(n) = vecsum([n/f[1]*f[2]|f<-factor(n+!n)~]); \\ A003415
a(n) = sumdiv(n, d, ad(d)*moebius(d)^2); \\ Michel Marcus, Oct 10 2021

Crossrefs

Cf. A003415 (arithmetic derivative), A008683 (mu), A008966.

Sequence in context: A271498 A365332 A367483 * A317940 A318674 A284118

Adjacent sequences: A348278 A348279 A348280 * A348282 A348283 A348284

Keyword

nonn

Author

Wesley Ivan Hurt, Oct 09 2021

Status

approved