Search: id:A013928
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%I A013928
%S A013928 0,1,2,3,3,4,5,6,6,6,7,8,8,9,10,11,11,12,12,13,13,14,15,16,16,16,17,17,
%T A013928 17,18,19,20,20,21,22,23,23,24,25,26,26,27,28,29,29,29,30,31,31,31,31,
               32,
%U A013928 32,33,33,34,34,35,36,37,37,38,39,39,39,40,41,42,42,43,44,45,45,46,47,
               47
%N A013928 Number of square-free numbers < n.
%C A013928 For n>=1 define an n X n (0,1) matrix A by A[i,j] = 1 if GCD(i,j) = 1, 
               A[i,j] = 0 if GCD(i,j) > 1 for 1<= i,j <=n . The rank of A is a(n+1) 
               . Asymptotic expression for a(n) is a(n) ~ n * 6 / Pi^2 - Sharon 
               Sela (sharonsela(AT)hotmail.com), May 06 2002
%H A013928 Daniel Forgues, <a href="b013928.txt">Table of n, a(n) for n=1..100000</
               a>
%H A013928 Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/
               Squarefree.html">Squarefree</a>.
%F A013928 a(n)=Sum_{k=1..n-1} mu(k)^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), 
               May 18 2001
%F A013928 a(n)=Sum_{d=1..floor(sqrt(n-1)} mu(d)*floor((n-1)/d^2), mu(d) = Moebius 
               function A008683. - Vladeta Jovovic (vladeta(AT)eunet.rs), Apr 06 
               2001
%F A013928 Asymptotic formula (with error term): a(n)=Sum_{k=1..n-1} mu(k)^2 = Sum_{k=1..n-1} 
               |mu(k)| = 6*n/Pi^2 + O(sqrt(n)) - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), 
               Jul 20 2002
%F A013928 a(n)=sum{k=0..n, if(k<=n-1, (mu(n-k) mod 2), 0)}; a(n+1)=sum{k=0..n, 
               mu(n-k+1) mod 2}; - Paul Barry (pbarry(AT)wit.ie), May 10 2005
%F A013928 a(n+1)=sum{k=0..n, abs(mu(n-k+1))}; - Paul Barry (pbarry(AT)wit.ie), 
               Jul 20 2005
%F A013928 a(n)=sum(k=1,floor(sqrt(n)),mu(k)*floor(n/k^2)) [From Benoit Cloitre 
               (benoit7848c(AT)orange.fr), Oct 25 2009]
%o A013928 (PARI) a(n)=sum(i=1,n,if(issquarefree(i),1,0))
%o A013928 (PARI) a(n)=sum(k=1,sqrtint(n),moebius(k)*floor(n/k^2)) [From Benoit 
               Cloitre (benoit7848c(AT)orange.fr), Oct 25 2009]
%Y A013928 Cf. A005117.
%Y A013928 Cf. A158819 Number of square-free numbers <= n minus round(n/zeta(2)). 
               [From Daniel Forgues (squid(AT)zensearch.com), May 26 2009]
%Y A013928 Sequence in context: A064047 A111899 A074753 this_sequence A006161 A132351 
               A025556
%Y A013928 Adjacent sequences: A013925 A013926 A013927 this_sequence A013929 A013930 
               A013931
%K A013928 nonn
%O A013928 1,3
%A A013928 Henri Lifchitz (100637.64(AT)CompuServe.COM)

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