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A053818 Sum_{k=1..n, gcd(n,k) = 1} k^2. 20
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

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 - A179880, A179882 - A179887, A179890, A179891, A007645, A003627, A034934. [From Jaroslav Krizek, Aug 01 2010]

REFERENCES

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.

LINKS

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.

FORMULA

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]

MAPLE

A053818 := proc(n)

    local a, k;

    a := 0 ;

    for k from 1 to n do

        if igcd(k, n) = 1 then

            a := a+k^2 ;

        end if;

    end do:

    a ;

end proc: # R. J. Mathar, Sep 26 2013

MATHEMATICA

f[n_] := Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2); Array[f, 43] [From Robert G. Wilson v, Jul 01 2010]

CROSSREFS

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

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Apr 07 2000

STATUS

approved

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Last modified December 21 02:15 EST 2014. Contains 252291 sequences.