A293227 a(n) is the number of proper divisors of n that are squarefree.

0, 1, 1, 2, 1, 3, 1, 2, 2, 3, 1, 4, 1, 3, 3, 2, 1, 4, 1, 4, 3, 3, 1, 4, 2, 3, 2, 4, 1, 7, 1, 2, 3, 3, 3, 4, 1, 3, 3, 4, 1, 7, 1, 4, 4, 3, 1, 4, 2, 4, 3, 4, 1, 4, 3, 4, 3, 3, 1, 8, 1, 3, 4, 2, 3, 7, 1, 4, 3, 7, 1, 4, 1, 3, 4, 4, 3, 7, 1, 4, 2, 3, 1, 8, 3, 3, 3, 4, 1, 8, 3, 4, 3, 3, 3, 4, 1, 4, 4, 4, 1, 7, 1, 4, 7

Offset: 1

Author

Antti Karttunen, Oct 08 2017

Keywords

• nonn

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A001221, A005117, A008683, A008966, A034444, A293228, A293234.

Cf. A001620, A306016.

Programs

Maple

with(numtheory): seq(coeff(series(add(mobius(k)^2*x^(2*k)/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 120); # Muniru A Asiru, Oct 28 2018

Mathematica

Table[Count[Most[Divisors[n]],?SquareFreeQ],{n,110}] (* _Harvey P. Dale, Jun 15 2021 *)
a[n_] := 2^PrimeNu[n] - Boole[SquareFreeQ[n]]; Array[a, 100] (* Amiram Eldar, Jun 30 2022 *)

PARI

A293227(n) = sumdiv(n, d, (d
				

Formula

a(n) = Sum_{d|n, dA008966(d).

a(n) = A034444(n) - A008966(n).

a(n) = 2^A001221(n) - A008683(n)^2 = 2^omega(n) - mu(n)^2.

G.f.: Sum_{k>=1} mu(k)^2*x^(2*k)/(1 - x^k). - Ilya Gutkovskiy, Oct 28 2018

Sum_{k=1..n} a(k) ~ (6/Pi^2)*n*(log(n) + 2*(gamma - 1 - zeta'(2)/zeta(2))), where gamma is Euler's constant (A001620). - Amiram Eldar, Dec 01 2023