A369700 Möbius transform of reduced totient function (A002322).

1, 0, 1, 1, 3, 0, 5, 0, 4, 0, 9, -1, 11, 0, -1, 2, 15, 0, 17, -1, -1, 0, 21, 0, 16, 0, 12, -1, 27, 0, 29, 4, -1, 0, 3, 0, 35, 0, -1, 0, 39, 0, 41, -1, 4, 0, 45, 0, 36, 0, -1, -1, 51, 0, 7, 0, -1, 0, 57, 1, 59, 0, -4, 8, -3, 0, 65, -1, -1, 0

Offset: 1

Author

Miles Englezou, Jan 29 2024

Keywords

• sign

Comments

Since A002322(n) = A000010(n) for n = 1, 2, 4, and odd prime powers, a(n) = A007431(n) for the same values of n.

Examples

			a(8) = mu(1)*lambda(8) + mu(2)*lambda(4) + mu(4)*lambda(2) + mu(8)*lambda(1) = 0.
		

Crossrefs

Cf. A008683, A002322, A000010, A007431.

Programs

Mathematica

a[n_] := DivisorSum[n, MoebiusMu[#] * CarmichaelLambda[n/#] &]; Array[a, 100] (* Amiram Eldar, Jan 29 2024 *)

PARI

a(n) = sumdiv(n, d, moebius(d)*lcm(znstar(n/d)[2]))

Formula

a(n) = Sum_{d|n} A008683(d) * A002322(n/d).