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 A053818 a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^2. 25
 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. - 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. - Jaroslav Krizek, Aug 01 2010 n does not divide a(n) iff n is a term in A316860, that is, either n is a power of 2 or n is a multiple of 3 and no prime factor of n is congruent to 1 mod 3. - Jianing Song, Jul 16 2018 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 G. C. Greubel, Table of n, a(n) for n = 1..2500 John D. Baum, A Number-Theoretic Sum, Mathematics Magazine 55.2 (1982): 111-113. P. G. Brown, Some comments on inverse arithmetic functions, Math. Gaz. 89 (516) (2005) 403-408. Constantin M. Petridi, The Sums of the k-powers of the Euler set and their connection with Artin's conjecture for primitive roots, arXiv:1612.07632 [math.NT], 2016. 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. - 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 a[n_] := Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2); Array[a, 43] (* Robert G. Wilson v, Jul 01 2010 *) PROG (PARI) a(n) = sum(k=1, n, k^2*(gcd(n, k) == 1)); \\ Michel Marcus, Jan 30 2016 CROSSREFS Cf. A023896, A053819, A053820. Cf. A143612. - Gary W. Adamson, Aug 27 2008 Cf. A179871-A179880, A179882-A179887, A179890, A179891, A007645, A003627, A034934. - Jaroslav Krizek, Aug 01 2010 Sequence in context: A105505 A005514 A069921 * A294286 A133629 A156302 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 October 23 14:54 EDT 2019. Contains 328345 sequences. (Running on oeis4.)