

A057627


Number of nonsquarefree numbers not exceeding n.


20



0, 0, 0, 1, 1, 1, 1, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 6, 6, 7, 7, 7, 7, 8, 9, 9, 10, 11, 11, 11, 11, 12, 12, 12, 12, 13, 13, 13, 13, 14, 14, 14, 14, 15, 16, 16, 16, 17, 18, 19, 19, 20, 20, 21, 21, 22, 22, 22, 22, 23, 23, 23, 24, 25, 25, 25, 25, 26, 26, 26, 26, 27, 27, 27, 28, 29, 29, 29
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,8


COMMENTS

Number of integers k in A013929 in the range 1 <= k <= n.
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(Moebius(k)^2, k=1..n).
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}. This sequence is different from A013940, albeit the first 35 terms are identical.


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

(Scheme) (define (A057627 n) ( n (A013928 (+ n 1))))
(PARI) a(n)=my(s); forprime(p=2, sqrtint(n), s+=n\p^2); s \\ Charles R Greathouse IV, May 18 2015


CROSSREFS

Cf. A005117, A008683, A013929, A013940, A013928, A002321, A028442, A107078 , A229099, A294242.
Sequence in context: A106742 A106733 A087838 * A013940 A029129 A087842
Adjacent sequences: A057624 A057625 A057626 * A057628 A057629 A057630


KEYWORD

nonn,easy


AUTHOR

Labos Elemer, Oct 10 2000


EXTENSIONS

Offset and formula corrected by Antti Karttunen, Jun 03 2014


STATUS

approved



