A383575 Characteristic function of numbers of the form k = m^e, where m is squarefree and (-1)^omega(k) = mu(e).

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

Offset

1

Comments

Characteristic function of A383016.

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) = A383576(n) + A382883(n).

Dirichlet g.f.: 1 + (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])})):
isA383575 := n -> n = 1 or(isA384667(n) and isA385055(n)):
A383575 := n -> ifelse(isA383575(n), 1, 0):
seq(A383575(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, A383016, A383576, A384667, A385055.

Sequence in context: A267612 A242252 A100810 * A385055 A373481 A174889

Adjacent sequences: A383572 A383573 A383574 * A383576 A383577 A383578

Keyword

nonn

Author

Friedjof Tellkamp, Jun 14 2025

Status

approved