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A053818
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Sum_{k=1..n, gcd(n,k) = 1} k^2.
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19
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1, 1, 5, 10, 30, 26, 91, 84, 159, 140, 385, 196, 650, 406, 620, 680, 1496, 654, 2109, 1080, 1806, 1650, 3795, 1544, 4150, 2756, 4365, 3164, 7714, 2360, 9455, 5456, 7370, 6256, 9940, 5196, 16206, 8778, 12324, 8560, 22140, 6972, 25585
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OFFSET
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1,3
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COMMENTS
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Equals row sums of triangle A143612 [From Gary W. Adamson, Aug 27 2008]
a(n) = A175505(n) * A023896(n) / A175506(n). For number n >= 1 holds B(n) = a(n) / A023896(n) = A175505(n) / A175506(n), where B(n) = antiharmonic mean of numbers k such that GCD(k, n) = 1 for k < n. Cf. A179871, A179872, A179873, A179874, A179875, A179876, A179877, A179878, A179879, A179880, A179882, A179883, A179884, A179885, A179886, A179887, A179890, A179891, A007645, A003627, A034934. [From Jaroslav Krizek, Aug 01 2010]
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REFERENCES
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T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 15, the function phi_2(n).
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #2.
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LINKS
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Table of n, a(n) for n=1..43.
P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408.
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FORMULA
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If n = p_1^e_1 * ... *p_r^e_r then a(n) = n^2*phi(n)/3 + (-1)^r*p_1*..._p_r*phi(n)/6.
a(n) = n^2*A000010(n)/3 + n*A023900(n)/6, n>1. [Brown]
a(n) = A000010(n)/3 * (n^2 + (-1)^A001221(n)*A007947(n)/2)) for n>=2. [From Jaroslav Krizek, Aug 24 2010]
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MATHEMATICA
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f[n_] := Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2); Array[f, 43] [From Robert G. Wilson v, Jul 01 2010]
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CROSSREFS
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Cf. A023896, A053819, A053820.
A143612 [From Gary W. Adamson, Aug 27 2008]
Sequence in context: A105505 A005514 A069921 * A133629 A156302 A156234
Adjacent sequences: A053815 A053816 A053817 * A053819 A053820 A053821
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KEYWORD
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nonn
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AUTHOR
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N. J. A. Sloane, Apr 07 2000
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STATUS
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approved
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