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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 (list; graph; refs; listen; history; text; internal format)
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^(p-1)+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

CROSSREFS

Cf. A005117, A071172, A053462.

Sequence in context: A186546 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

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Last modified June 24 05:50 EDT 2017. Contains 288697 sequences.