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Search: id:A013928
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| A013928 |
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Number of square-free numbers < n. |
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+0
19
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| 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, 17, 18, 19, 20, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29, 29, 29, 30, 31, 31, 31, 31, 32, 32, 33, 33, 34, 34, 35, 36, 37, 37, 38, 39, 39, 39, 40, 41, 42, 42, 43, 44, 45, 45, 46, 47, 47
(list; graph; listen)
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OFFSET
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1,3
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COMMENT
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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
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LINKS
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Daniel Forgues, Table of n, a(n) for n=1..100000
Eric Weisstein's World of Mathematics, Squarefree.
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FORMULA
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a(n)=Sum_{k=1..n-1} mu(k)^2. - Vladeta Jovovic (vladeta(AT)eunet.rs), May 18 2001
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
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
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
a(n+1)=sum{k=0..n, abs(mu(n-k+1))}; - Paul Barry (pbarry(AT)wit.ie), Jul 20 2005
a(n)=sum(k=1,floor(sqrt(n)),mu(k)*floor(n/k^2)) [From Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 25 2009]
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PROGRAM
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(PARI) a(n)=sum(i=1, n, if(issquarefree(i), 1, 0))
(PARI) a(n)=sum(k=1, sqrtint(n), moebius(k)*floor(n/k^2)) [From Benoit Cloitre (benoit7848c(AT)orange.fr), Oct 25 2009]
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CROSSREFS
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Cf. A005117.
Cf. A158819 Number of square-free numbers <= n minus round(n/zeta(2)). [From Daniel Forgues (squid(AT)zensearch.com), May 26 2009]
Sequence in context: A064047 A111899 A074753 this_sequence A006161 A132351 A025556
Adjacent sequences: A013925 A013926 A013927 this_sequence A013929 A013930 A013931
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KEYWORD
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nonn
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AUTHOR
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Henri Lifchitz (100637.64(AT)CompuServe.COM)
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