A370598 - OEIS

%I #9 Feb 23 2024 08:29:51

%S 1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%T 1,1,1,1,1,1,1,1,1,1,1,1,1,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,

%U 1,1,1,1,1,1,1,1,1,1,1,0,0,1,1,1,1,1,1

%N Characteristic function of exponentially squarefree numbers (A209061).

%H Amiram Eldar, <a href="/A370598/b370598.txt">Table of n, a(n) for n = 1..10000</a>

%H Jean-Marie De Koninck and William Verreault, <a href="https://arxiv.org/abs/2308.09149">On the tower factorization of integers</a>, arXiv:2308.09149 [math.NT], 2023. See page 2.

%H <a href="/index/Ch#char_fns">Index entries for sequences related to characteristic functions</a>.

%F Multiplicative with a(p^e) = mu(e)^2, where mu is the Möbius function (A008683).

%F a(n) = 1 if and only if n is in A209061.

%F a(n) = 0 if and only if n is in A130897.

%F a(n) = abs(A166234(n)).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = A262276.

%t f[p_, e_] := MoebiusMu[e]^2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = vecprod(apply(x -> moebius(x)^2, factor(n)[, 2]));

%Y Cf. A008683, A130897, A209061, A262276.

%K nonn,easy,mult

%O 1

%A _Amiram Eldar_, Feb 23 2024