A389204 a(n) = A325973(n) - A325974(n), where A325973 and A325974 are arithmetic means of {sum of unitary divisors and sum of squarefree divisors} and of {sum of non-unitary divisors and sum of nonsquarefree divisors}, respectively.

1, 3, 4, 1, 6, 12, 8, -3, 1, 18, 12, 4, 14, 24, 24, -11, 18, 3, 20, 6, 32, 36, 24, -12, 1, 42, -8, 8, 30, 72, 32, -27, 48, 54, 48, -29, 38, 60, 56, -18, 42, 96, 44, 12, 6, 72, 48, -44, 1, 3, 72, 14, 54, -24, 72, -24, 80, 90, 60, 24, 62, 96, 8, -59, 84, 144, 68, 18, 96, 144, 72, -93, 74, 114, 4, 20, 96, 168, 80, -66

Offset

1,2

Links

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

Formula

a(n) = A325973(n) - A325974(n).

a(n) = A000203(n) - 2*A325974(n) = 2*A325973(n) - A000203(n).

From Amiram Eldar, Oct 06 2025: (Start)

a(n) = A034448(n) + A048250(n) - A000203(n).

Dirichlet g.f.: zeta(s)*zeta(s-1)*(1/zeta(2*s-1) + 1/zeta(2*s-2) - 1).

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = 1 - zeta(2) * (1 - 1/zeta(3)) = 0.72349871... . (End)

Mathematica

f1[p_, e_] := (p^(e+1)-1)/(p-1); f2[p_, e_] := p + 1; a[n_] := Module[{f = FactorInteger[n]}, Times @@ (1 + Power @@@ f) - Times @@ f1 @@@ f + Times @@ f2 @@@ f]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Oct 06 2025 *)

Prog

(PARI)
A325973(n) = (1/2)*sumdiv(n, d, d*(issquarefree(d) + (1==gcd(d, n/d))));
A325974(n) = (1/2)*sumdiv(n, d, d*((1-issquarefree(d))+(gcd(d, n/d)>1)));
A389204(n) = (A325973(n)-A325974(n));

Crossrefs

Cf. A000203, A034448, A048250, A325973, A325974.

Cf. A028982 (positions of odd terms), A048107 (of positive terms), A048108 (of negative terms), A191217 (of terms of the form 4u+2), A280076 (of 1's, conjectured).

Cf. also A389205, A389210.

Sequence in context: A348503 A348047 A348984 * A323159 A347090 A328181

Adjacent sequences: A389201 A389202 A389203 * A389205 A389206 A389207

Keyword

sign,easy

Author

Antti Karttunen, Oct 06 2025

Status

approved