

A143658


Number of squarefree integers not exceeding 2^n.


7



1, 2, 3, 6, 11, 20, 39, 78, 157, 314, 624, 1245, 2491, 4982, 9962, 19920, 39844, 79688, 159360, 318725, 637461, 1274918, 2549834, 5099650, 10199301, 20398664, 40797327, 81594626, 163189197, 326378284, 652756722, 1305513583, 2611027094
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OFFSET

0,2


COMMENTS

Except for the first 2 terms, it would not make a difference to replace "not exceeding" by "less than": that sequence would start 0,1,3,6,11,20,39,78,...


LINKS

Peter Polm & Gerard P. Michon, Table of n, a(n) for n = 0..64 (terms up to n=58 from Gerard P. Michon)
Project Euler, Problem 193: Squarefree Numbers
G. P. Michon, On the number of squarefree integers not exceeding N.  Gerard P. Michon, Apr 30 2009


FORMULA

a(n) = Sum for i = 1 to 2^(n/2) of A008683(i)*floor(2^n/i^2).  Gerard P. Michon, Apr 30 2009
The limit of a(n)/2^n is 6/Pi^2.  Gerard P. Michon, Apr 30 2009


EXAMPLE

a(4) = 11 since there are the 11 squarefree integers {1, 2, 3, 5, 6, 7, 10, 11, 13, 14, 15} not exceeding 2^4=16.


MATHEMATICA

c = 0; k = 1; lst = {1}; Do[ While[k <= 2^n, If[ SquareFreeQ@k, c++ ]; k++ ]; AppendTo[lst, c], {n, 27}] (* Robert G. Wilson v, Aug 31 2008 *)


PROG

(PARI) print1(s=1); for(p=1, 20, print1(", ", s+=sum(k=2^(p1)+1, 2^p, issquarefree(k))))
(PARI) a(n)=sum(d=1, sqrtint(n=2^n), moebius(d)*n\d^2) \\ Charles R Greathouse IV, Nov 14 2012
(PARI) a(n)=my(s); forsquarefree(d=1, sqrtint(n=2^n), s += n\d[1]^2*moebius(d)); s \\ Charles R Greathouse IV, Jan 08 2018


CROSSREFS

Cf. A005117, A071172, A053462.
Sequence in context: A318910 A141435 A096080 * A005230 A030037 A077078
Adjacent sequences: A143655 A143656 A143657 * A143659 A143660 A143661


KEYWORD

nonn


AUTHOR

M. F. Hasler, Aug 28 2008


EXTENSIONS

5 more terms from Robert G. Wilson v, Aug 31 2008
More terms from Alexis Olson (AlexisOlson(AT)gmail.com), Nov 08 2008


STATUS

approved



