A304362 a(n) = Sum_{d|n, d = 1 or not a perfect power} mu(n/d).

1, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, -1, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, -1, 0, 1, 0, 0, 0, 0, 0, 1, 1, -1, 0, 0, 0, 0, 0

Offset

1

Comments

The Moebius function mu is defined by mu(n) = (-1)^k if n is a product of k distinct primes and mu(n) = 0 otherwise.

Up to n = 10^7 this sequence only takes values in {-2, -1, 0, 1, 2}. Is this true in general?

Links

Antti Karttunen, Table of n, a(n) for n = 1..65537

Formula

a(n) = mu(n) + Sum_{d * e = n, d in A007916, e in A005117} (-1)^omega(e), where mu = A008683 and omega = A001221.

Mathematica

Table[Sum[If[GCD@@FactorInteger[d][[All, 2]]===1, MoebiusMu[n/d], 0], {d, Divisors[n]}], {n, 100}]

Prog

(PARI) A304362(n) = sumdiv(n, d, if(!ispower(d), moebius(n/d), 0)); \\ Antti Karttunen, Jul 29 2018

Crossrefs

Cf. A000005, A000961, A001221, A001597, A001694, A005117, A007916, A008683, A091050, A203025, A304326, A304327, A304364, A304365, A304369.

Sequence in context: A353800 A353374 A326072 * A368997 A330682 A230135

Adjacent sequences: A304359 A304360 A304361 * A304363 A304364 A304365

Keyword

sign

Author

Gus Wiseman, May 11 2018

Extensions

More terms from Antti Karttunen, Jul 29 2018

Status

approved