A074722 a(n) = Sum_{d divides n} phi(n/d)*(-1)^bigomega(d).

1, 0, 1, 2, 3, 0, 5, 2, 5, 0, 9, 2, 11, 0, 3, 6, 15, 0, 17, 6, 5, 0, 21, 2, 17, 0, 13, 10, 27, 0, 29, 10, 9, 0, 15, 10, 35, 0, 11, 6, 39, 0, 41, 18, 15, 0, 45, 6, 37, 0, 15, 22, 51, 0, 27, 10, 17, 0, 57, 6, 59, 0, 25, 22, 33, 0, 65, 30, 21, 0, 69, 10, 71, 0, 17, 34, 45, 0, 77, 18, 41, 0

Offset: 1

Author

Vladeta Jovovic, Sep 27 2002

Keywords

• nonn
• easy
• mult

Comments

a(n) = 0 if and only if n == 2 (mod 4). - Robert Israel, Jan 04 2017

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000010, A001222, A008683, A058026, A206369.

Programs

Maple

f:= proc(n) uses numtheory; local d;
   add(phi(n/d)*(-1)^bigomega(d), d=divisors(n))
end proc:
map(f, [$1..100]); # Robert Israel, Jan 04 2017

Mathematica

f[d_] := EulerPhi[n/d] LiouvilleLambda[d]
Table[DivisorSum[n, f], {n, 1, 50}] (* Benedict W. J. Irwin, Jul 11 2018 *)
f[p_, e_] := 2*(-1)^(e + 1)*((-p)^(e + 1) - 1)/(p + 1) - p^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 30 2022 *)

PARI

a(n) = sumdiv(n, d, eulerphi(n/d)*(-1)^bigomega(d)); \\ Michel Marcus, Jul 11 2018

Formula

Multiplicative with a(p^e) = 2*(-1)^(e+1)*((-p)^(e+1)-1)/(p+1)-p^e.

Dirichlet g.f.: zeta(2s)*zeta(s-1)/(zeta(s)^2). - Benedict W. J. Irwin, Jul 11 2018

Sum_{k=1..n} a(k) ~ n^2 / 5. - Vaclav Kotesovec, Feb 01 2019

a(n) = Sum_{k=1..n} (-1)^bigomega(gcd(n,k)). - Ilya Gutkovskiy, Feb 22 2020

Möbius transform of A206369: a(n) = Sum_{d|n} A008683(d) * A206369(n/d). - Amiram Eldar, Aug 28 2023