A158819 (Number of squarefree numbers <= n) minus round(n/zeta(2)).

0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 1, 2, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 2, 1, 2, 1, 1, 2, 2, 2, 2, 1, 1, 0, 1, 1, 1, 1, 1

Offset

1,7

Comments

Race between the number of squarefree numbers and round(n/zeta(2)).

First term < 0: a(172) = -1.

References

G. H. Hardy and S. Ramanujan, The normal number of prime factors of a number n, Q. J. Math., 48 (1917), pp. 76-92.

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth edition (1979), Clarendon Press, pp. 269-270.

Links

Daniel Forgues, Table of n, a(n) for n=1..100000

A. Granville, ABC means we can count squarefree

Formula

Since zeta(2) = Sum_{i>=1} 1/(i^2) = (Pi^2)/6, we get:

a(n) = A013928(n+1) - n/Sum_{i>=1} 1/(i^2) = O(sqrt(n));

a(n) = A013928(n+1) - 6*n/(Pi^2) = O(sqrt(n)).

Crossrefs

Cf. A008966 1 if n is squarefree, else 0.

Cf. A013928 Number of squarefree numbers < n.

Cf. A100112 If n is the k-th squarefree number then k else 0.

Cf. A057627 Number of nonsquarefree numbers not exceeding n.

Cf. A005117 Squarefree numbers.

Cf. A013929 Nonsquarefree numbers.

Sequence in context: A357900 A357732 A356428 * A031279 A124778 A037831

Adjacent sequences: A158816 A158817 A158818 * A158820 A158821 A158822

Keyword

sign

Author

Daniel Forgues, Mar 27 2009

Status

approved