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

1,3

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

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A106316 A300721 A126707 * A057458 A291193 A291200

Adjacent sequences: A326302 A326303 A326304 * A326306 A326307 A326308

Keyword

nonn,mult

Author

Ilya Gutkovskiy, Oct 17 2019

Status

approved