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

1,4

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from exponents in factorization of n.

Formula

a(n) = Sum_{d|n, d<n} A008966(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

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 *)

Prog

(PARI) A293227(n) = sumdiv(n, d, (d<n)*issquarefree(d));

Crossrefs

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

Cf. A001620, A306016.

Sequence in context: A182850 A323014 A350331 * A291208 A241165 A233183

Adjacent sequences: A293224 A293225 A293226 * A293228 A293229 A293230

Keyword

nonn

Author

Antti Karttunen, Oct 08 2017

Status

approved