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A018805
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Number of elements in the set {(x,y): 1<=x,y<=n, 1=gcd(x,y)}.
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28
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1, 3, 7, 11, 19, 23, 35, 43, 55, 63, 83, 91, 115, 127, 143, 159, 191, 203, 239, 255, 279, 299, 343, 359, 399, 423, 459, 483, 539, 555, 615, 647, 687, 719, 767, 791, 863, 899, 947, 979, 1059, 1083, 1167, 1207, 1255, 1299, 1391, 1423, 1507, 1547, 1611, 1659, 1763
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Equals partial sums of A140434 (1, 2, 4, 4, 8, 4, 12, 8,...) and row sums of triangle A143469. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 17 2008]
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REFERENCES
| Cai, Jin-Yi and Bach, Eric. On testing for zero polynomials by a set of points with bounded precision, Theoret. Comput. Sci. 296 (2003), no. 1, 15-25. MR1965515 (2004m:68279).
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 110-112.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954.
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LINKS
| Olivier GERARD, Table of n, a(n) for n = 1..100000 [Replaces an earlier b-file from Charles R Greathouse IV]
Pieter Moree, Counting carefree couples
Eric Weisstein's World of Mathematics, Carefree Couple
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FORMULA
| a(n) = 2*( Sum phi(j), j=1..n ) - 1.
a(n) = n^2 - Sum a([ n/j ]), j=2..n.
a(n) = 2*A015614(n) + 1. - Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 08 2006
a(n) = 2*A002088(n) - 1. - Hugo van der Sanden (hv(AT)crypt.org), Nov 22 2008
a(n) ~ (1/Zeta(2)) * n^2 = (6/pi^2) * n^2 as n goes to infinity (zeta is the Riemann zeta function, A013661, and the constant 6/pi^2 is 0.607927..., A059956). - Ahmed Fares (ahmedfares(AT)my-deja.com), Jul 18 2001
a(n) = sum(k=1, n, mu(k)*floor(n/k)^2). - Benoit Cloitre (benoit7848c(AT)orange.fr), May 11 2003
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MATHEMATICA
| FoldList[ Plus, 1, 2 Array[ EulerPhi, 60, 2 ] ]
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PROG
| (PARI) a(n)=sum(k=1, n, moebius(k)*(n\k)^2)
(PARI)
A018805(n)=2 *sum(j=1, n, eulerphi(j)) - 1;
for(n=1, 99, print1(A018805(n), ", ")); /* show terms */
(MAGMA) /* based on the first formula */ A018805:=func< n | 2*&+[ EulerPhi(k): k in [1..n] ]-1 >; [ A018805(n): n in [1..60] ]; // Klaus Brockhaus, Jan 27 2011
(MAGMA) /* based on the second formula */ A018805:=func< n | n eq 1 select 1 else n^2-&+[ $$(n div j): j in [2..n] ] >; [ A018805(n): n in [1..60] ]; // Klaus Brockhaus, Feb 07 2011
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CROSSREFS
| Cf. A015614, A002088, A100613 (gcd > 1), A071778 (triples).
Cf. A143469, A140434 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Aug 17 2008]
Cf. A013661, A059956.
Sequence in context: A092109 A117991 A118260 * A191037 A135932 A105876
Adjacent sequences: A018802 A018803 A018804 * A018806 A018807 A018808
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KEYWORD
| nonn
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AUTHOR
| David W. Wilson (davidwwilson(AT)comcast.net)
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EXTENSIONS
| Mathematica program Aug 15 1997 (Olivier Gerard).
More terms from Reinhard Zumkeller (reinhard.zumkeller(AT)gmail.com), Apr 08 2006
Link to Moree's paper corrected Peter Luschny (peter(AT)luschny.de), Aug 08 2009
b-file of first 100000 terms by Olivier GERARD Feb 06 2011
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