A181553 -id:A181553 - cp's OEIS frontend

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

Peter Luschny, Oct 30 2010

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).

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Peter Luschny, Sequences related to Euler's totient function.

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

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

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 *)

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

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)

A181552 T(n,k) = gcd(n,k) A181549(k), triangle read by rows.

Original entry on oeis.org

1, 1, 6, 1, 3, 12, 1, 6, 4, 20, 1, 3, 4, 5, 30, 1, 6, 12, 10, 6, 72, 1, 3, 4, 5, 6, 12, 56, 1, 6, 4, 20, 6, 24, 8, 80, 1, 3, 12, 5, 6, 36, 8, 10, 99, 1, 6, 4, 10, 30, 24, 8, 20, 11, 180, 1, 3, 4, 5, 6, 12, 8, 10, 11, 18, 132, 1, 6, 12, 20, 6, 72, 8, 40, 33, 36, 12, 240

Offset: 1

Peter Luschny, Oct 30 2010

A181549(n) = sum{k|n} k mu_2(n/k) is a variant of Euler's phi function relative to the Moebius function of order 2.

			1,
1,6,
1,3,12,
1,6,.4,20,
1,3,.4,.5,30,
1,6,12,10,.6,72,
1,3,.4,.5,.6,12,56,
1,6,.4,20,.6,24,.8,80,

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

Cf. A130212, A181538, row sums of triangle is A181553.

A181552 := (n,k) -> igcd(n,k)*A181549(k);

mu2[1] = 1; mu2[n_] := Sum[Boole[Divisible[n, d^2]]*MoebiusMu[n/d^2]*MoebiusMu[n/d], {d, Divisors[n]}]; A181549[n_] := Sum[k*mu2[n/k], {k, Divisors[n]}]; t[n_, k_] := GCD[n, k]*A181549[k]; Table[t[n, k], {n, 1, 12}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 05 2014 *)

cp's OEIS Frontend

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

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