A326305 Dirichlet g.f.: zeta(s-1) * (1 - 2^(-s)) / zeta(s).

1, 0, 2, 1, 4, 0, 6, 2, 6, 0, 10, 2, 12, 0, 8, 4, 16, 0, 18, 4, 12, 0, 22, 4, 20, 0, 18, 6, 28, 0, 30, 8, 20, 0, 24, 6, 36, 0, 24, 8, 40, 0, 42, 10, 24, 0, 46, 8, 42, 0, 32, 12, 52, 0, 40, 12, 36, 0, 58, 8, 60, 0, 36, 16, 48, 0, 66, 16, 44, 0, 70, 12, 72, 0, 40

Offset: 1

Author

Ilya Gutkovskiy, Oct 17 2019

Keywords

• nonn
• mult

Comments

Moebius transform of A026741.

Dirichlet convolution of A002131 with Dirichlet inverse of A000005.

Dirichlet convolution of A000027 with Dirichlet inverse of A001511.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000010, A000027, A001511, A002131, A007427, A008683, A016825 (positions of 0's), A026741, A092673, A299069.

Programs

Magma

[IsOdd(n) select EulerPhi(n) else EulerPhi(n)-EulerPhi(n div 2) : n in [1..80]]; // Marius A. Burtea, Oct 17 2019

Mathematica

Table[Sum[MoebiusMu[n/d] Numerator[d/2], {d, Divisors[n]}], {n, 1, 75}]
a[n_] := If[OddQ[n], EulerPhi[n], EulerPhi[n] - EulerPhi[n/2]]; Table[a[n], {n, 1, 75}]
f[2, e_] := If[e == 1, 0, 2^(e - 2)]; f[p_, e_] := (p - 1)*p^(e - 1); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Nov 30 2020 *)

Formula

a(n) = phi(n) if n odd, phi(n) - phi(n/2) if n even, where phi = A000010.

a(n) = Sum_{d|n} mu(n/d) * A026741(d).

a(n) = Sum_{d|n} A007427(n/d) * A002131(d).

a(n) = Sum_{d|n} A092673(n/d) * d.

a(p) = p - 1, where p is odd prime.

Product_{n>=1} 1 / (1 - x^n)^a(n) = g.f. for A299069.

Sum_{k=1..n} a(k) ~ 9*n^2 / (4*Pi^2). - Vaclav Kotesovec, Oct 26 2019

Multiplicative with a(2^e) = 0 if e = 1 and 2^(e-2) otherwise, and a(p^e) = (p-1)*p^(e-1) for odd primes p. - Amiram Eldar, Nov 30 2020