A389162 a(n) = A034448(n) - A048250(n), where A034448 is the sum of unitary divisors and A048250 is the sum of squarefree divisors.

0, 0, 0, 2, 0, 0, 0, 6, 6, 0, 0, 8, 0, 0, 0, 14, 0, 18, 0, 12, 0, 0, 0, 24, 20, 0, 24, 16, 0, 0, 0, 30, 0, 0, 0, 38, 0, 0, 0, 36, 0, 0, 0, 24, 36, 0, 0, 56, 42, 60, 0, 28, 0, 72, 0, 48, 0, 0, 0, 48, 0, 0, 48, 62, 0, 0, 0, 36, 0, 0, 0, 78, 0, 0, 80, 40, 0, 0, 0, 84, 78, 0, 0, 64, 0, 0, 0, 72, 0, 108, 0, 48, 0, 0, 0, 120

Offset

1,4

Links

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

Formula

From Amiram Eldar, Oct 05 2025: (Start)

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

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = zeta(2)/zeta(3) - 1 = A306633 - 1. (End)

Mathematica

a[n_] := Module[{f = FactorInteger[n]}, Times @@ (1 + Power @@@ f) - Times @@ (f[[;; , 1]] + 1)]; Array[a, 100] (* Amiram Eldar, Oct 05 2025 *)

Prog

(PARI)
A034448(n) = { my(f=factorint(n)); prod(k=1, #f~, 1+(f[k, 1]^f[k, 2])); };
A048250(n) = factorback(apply(p -> p+1, factor(n)[, 1]));
A389162(n) = (A034448(n)-A048250(n));

Crossrefs

Cf. A034448, A048250, A306633, A323159, A325973, A389074, A389075.

Sequence in context: A334045 A028973 A066503 * A057385 A231108 A231285

Adjacent sequences: A389159 A389160 A389161 * A389163 A389164 A389165

Keyword

nonn,easy

Author

Antti Karttunen, Oct 04 2025

Status

approved