A254503 Möbius transform of A034448.

1, 2, 3, 2, 5, 6, 7, 4, 6, 10, 11, 6, 13, 14, 15, 8, 17, 12, 19, 10, 21, 22, 23, 12, 20, 26, 18, 14, 29, 30, 31, 16, 33, 34, 35, 12, 37, 38, 39, 20, 41, 42, 43, 22, 30, 46, 47, 24, 42, 40, 51, 26, 53, 36, 55, 28, 57, 58, 59, 30, 61, 62, 42, 32, 65, 66, 67, 34, 69, 70

Offset: 1

Author

Álvar Ibeas, Jan 31 2015

Keywords

• mult
• nonn
• easy

Links

• Álvar Ibeas, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010 (totient), A001694 (powerful), A005117 (squarefree), A034448 (usigma), A057521 (powerful part), A055231 (unitary squarefree kernel).

Programs

Mathematica

Table[DivisorSum[n, MoebiusMu[#]^2*EulerPhi[n/#] &, CoprimeQ[n/#, #] &], {n, 70}] (* Michael De Vlieger, Jun 27 2018 *)
f[p_, e_] := (p - 1)*p^(e - 1); f[p_, 1] := p; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

PARI

a(n) = {my(f = factor(n)); for (i=1, #f~, if ((e=f[i, 2]) > 1, f[i, 1] = eulerphi(f[i, 1]^e); f[i, 2] = 1);); factorback(f);} \\ Michel Marcus, Feb 06 2015

PARI

a(n) = sumdiv(n, d, if(gcd(n/d, d) == 1, moebius(d)^2 * eulerphi(n/d))); \\ Daniel Suteu, Jun 27 2018

Formula

a(n) = phi(A057521(n)) * A055231(n).

If n is squarefree, a(n) = n; if n is powerful, a(n) = phi(n).

Multiplicative with a(p) = p; a(p^e) = phi(p^e), for e > 1.

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

a(n) = Sum_{d|n, gcd(n/d, d) = 1} mu(d)^2 * phi(n/d). - Daniel Suteu, Jun 27 2018

Sum_{k=1..n} a(k) ~ n^2 / (2*zeta(3)). - Vaclav Kotesovec, Jan 11 2019