A383576 Characteristic function of numbers of the form k = m^e, where m is squarefree and (-1)^omega(k) = -mu(e).

0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0

Offset

1

Comments

Characteristic function of A383017.

a(n) = 0 if any two exponents in the prime factorization of n differ.

It follows from the definition that e is also squarefree.

Links

Friedjof Tellkamp, Table of n, a(n) for n = 1..10000

Formula

a(n) = A383575(n) - A382883(n).

Dirichlet g.f.: (1/2) * Sum_{k>=1} mu(k)^2 * (zeta(k*s)/zeta(2*k*s) - 1) - mu(k) * (1/zeta(k*s) - 1).

Maple

isA384667 := n -> local p; is(1 >= nops({seq(p[2], p in ifactors(n)[2])})):
isA383576 := n -> n > 1 and isA384667(n) and isA384709(n):
A383576 := n -> ifelse(isA383576(n), 1, 0):
seq(A383576(n), n = 1..86);  # Peter Luschny, Jun 16 2025

Mathematica

a[n_] := Boole[fi = FactorInteger[n]; Equal@@fi[[All, 2]] && (-1)^PrimeNu[n] == -MoebiusMu[fi[[1, 2]]]]; Array[a, 86]

Crossrefs

Cf. A001221, A005117, A008683, A382883, A383017, A383575, A384667, A384709.

Sequence in context: A252233 A374130 A374466 * A283991 A353499 A359372

Adjacent sequences: A383573 A383574 A383575 * A383577 A383578 A383579

Keyword

nonn

Author

Friedjof Tellkamp, Jun 14 2025

Status

approved