A285879 Number of odd squarefree numbers <= n.

1, 1, 2, 2, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 7, 8, 8, 9, 9, 10, 10, 11, 11, 11, 11, 11, 11, 12, 12, 13, 13, 14, 14, 15, 15, 16, 16, 17, 17, 18, 18, 19, 19, 19, 19, 20, 20, 20, 20, 21, 21, 22, 22, 23, 23, 24, 24, 25, 25, 26, 26, 26, 26, 27, 27, 28, 28, 29, 29, 30, 30, 31, 31, 31, 31, 32, 32, 33, 33, 33, 33, 34, 34, 35

Offset

1,3

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Squarefree

Formula

G.f.: Sum_{k>=1} x^A056911(k)/(1 - x).

a(n) ~ 4*n/Pi^2. See A185199.

From Ridouane Oudra, Apr 17 2026: (Start)

a(n) = Sum_{k=1..n} A323239(k).

a(n) = Sum_{k=1..n} A359548(k)*floor(n/k).

a(n) = Sum_{k=1..floor(sqrt(n))} A087003(k)*round(n/(2*k^2)).

a(n) = A013928(n+1) - A285881(n).

a(n) = A285881(2*n). (End)

Maple

ListTools:-PartialSums(map(op, [seq(`if`(numtheory:-issqrfree(n), [1, 0], [0, 0]), n=1..100, 2)])); # Robert Israel, May 07 2018
# Alternative:
seq(add(numtheory[mobius](2*k)^2, k=1..n), n=1..120); # Ridouane Oudra, Apr 17 2026

Mathematica

Table[Sum[Boole[OddQ[k] && SquareFreeQ[k]], {k, 1, n}], {n, 85}]
nmax = 85; Rest[CoefficientList[Series[Sum[Boole[OddQ[k] && MoebiusMu[k]^2 == 1] x^k/(1 - x), {k, 1, nmax}], {x, 0, nmax}], x]]

Prog

(PARI) a(n) = sum(k=1, n, (k%2)*issquarefree(k)); \\ Michel Marcus, Apr 27 2017
(Python)
from sympy.ntheory.factor_ import core
def a(n): return sum([1 for k in range(1, n + 1) if k%2==1 and core(k)==k]) # Indranil Ghosh, Apr 28 2017
(Python)
from math import isqrt
from sympy import mobius
def A285879(n): return int(sum(mobius(k)*(n//k**2+1>>1) for k in range(1, isqrt(n)+1, 2))) # Chai Wah Wu, Apr 08 2026

Crossrefs

Cf. A005117, A008683, A013928, A056911, A185199, A285881, A323239, A359548, A087003.

Sequence in context: A283992 A074796 A061070 * A117695 A375502 A074794

Adjacent sequences: A285876 A285877 A285878 * A285880 A285881 A285882

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 27 2017

Status

approved