A181549 a(n) = Sum_{k|n} k*mu_2(n/k).

0, 1, 3, 4, 5, 6, 12, 8, 10, 11, 18, 12, 20, 14, 24, 24, 20, 18, 33, 20, 30, 32, 36, 24, 40, 29, 42, 33, 40, 30, 72, 32, 40, 48, 54, 48, 55, 38, 60, 56, 60, 42, 96, 44, 60, 66, 72, 48, 80, 55, 87, 72, 70, 54, 99, 72, 80, 80, 90, 60

Offset

0,3

Comments

Sum_{k|n} k*mu(n/k) is Euler's phi function. In A181549 mu(n) is replaced by the Moebius function of order 2, mu_2(n), A189021(n).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Peter Luschny, Sequences related to Euler's totient function.

Formula

From Amiram Eldar, Nov 30 2022: (Start)

Multiplicative with a(p)= p + 1, and a(p^e) = p^e + p^(e-1) - p^(e-2) if e > 1.

Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 + 1/p^2 - 1/p^4) = 0.7124102278... . (End)

Maple

A181549 := proc(n) local k; add(k*A189021(n/k), k=divisors(n)) end;

Mathematica

mu2[1] = 1; mu2[n_] := Sum[Boole[Divisible[n, d^2]]*MoebiusMu[n/d^2]*MoebiusMu[n/d], {d, Divisors[n]}]; a[n_] := Sum[k*mu2[n/k], {k, Divisors[n]}]; Table[a[n], {n, 0, 59}] (* Jean-François Alcover, Feb 05 2014 *)
f[p_, e_] := p^e + p^(e - 1) - If[e > 1, p^(e - 2), 0]; a[0] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Nov 30 2022 *)

Prog

(PARI) a(n) = if(n == 0, 0, my(f = factor(n)); prod(i = 1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1) - if(f[i, 2] > 1, f[i, 1]^(f[i, 2]-2), 0))); \\ Amiram Eldar, Nov 30 2022

Crossrefs

Cf. A000010, A008683, A181550, A181552, A181553, A189021.

Sequence in context: A365172 A331107 A069184 * A366539 A366537 A241405

Adjacent sequences: A181546 A181547 A181548 * A181550 A181551 A181552

Keyword

nonn,mult

Author

Peter Luschny, Oct 30 2010

Status

approved