OFFSET
1,8
COMMENTS
Number of integers k in A013929 in the range 1 <= k <= n.
This sequence is different from A013940, albeit the first 35 terms are identical.
Asymptotic to k*n where k = 1 - 1/zeta(2) = 1 - 6/Pi^2 = A229099. - Daniel Forgues, Jan 28 2011
This sequence is the sequence of partial sums of A107078 (not of A056170). - Jason Kimberley, Feb 01 2017
Number of partitions of 2n into two parts with the smallest part nonsquarefree. - Wesley Ivan Hurt, Oct 25 2017
LINKS
Robert Israel, Table of n, a(n) for n = 1..10000
FORMULA
a(n) = n - A013928(n+1) = n - Sum_{k=1..n} mu(k)^2.
G.f.: Sum_{k>=1} (1 - mu(k)^2)*x^k/(1 - x). - Ilya Gutkovskiy, Apr 17 2017
EXAMPLE
a(36)=13 because 13 nonsquarefree numbers exist which do not exceed 36:{4,8,9,12,16,18,20,24,25,27,28,32,36}.
MAPLE
N:= 1000: # to get terms up to a(N)
B:= Array(1..N, numtheory:-issqrfree):
C:= map(`if`, B, 0, 1):
A:= map(round, Statistics:-CumulativeSum(C)):
seq(A[n], n=1..N); # Robert Israel, Jun 03 2014
MATHEMATICA
Accumulate[Table[If[SquareFreeQ[n], 0, 1], {n, 80}]] (* Harvey P. Dale, Jun 04 2014 *)
PROG
(PARI) a(n) = my(s=0); forsquarefree(k=1, sqrtint(n), s += (-1)^(#k[2]~) * (n\k[1]^2)); n - s; \\ Charles R Greathouse IV, May 18 2015; corrected by Daniel Suteu, May 11 2023
(Python)
from math import isqrt
from sympy import mobius
def A057627(n): return n-sum(mobius(k)*(n//k**2) for k in range(1, isqrt(n)+1)) # Chai Wah Wu, May 10 2024
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Labos Elemer, Oct 10 2000
EXTENSIONS
Offset and formula corrected by Antti Karttunen, Jun 03 2014
STATUS
approved