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%I #61 Dec 03 2023 05:03:14
%S 1,1,5,10,30,26,91,84,159,140,385,196,650,406,620,680,1496,654,2109,
%T 1080,1806,1650,3795,1544,4150,2756,4365,3164,7714,2360,9455,5456,
%U 7370,6256,9940,5196,16206,8778,12324,8560,22140,6972,25585
%N a(n) = Sum_{k=1..n, gcd(n,k) = 1} k^2.
%C Equals row sums of triangle A143612. - _Gary W. Adamson_, Aug 27 2008
%C 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
%C 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
%D Tom M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 48, problem 15, the function phi_2(n).
%D Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 199, #2.
%H G. C. Greubel, <a href="/A053818/b053818.txt">Table of n, a(n) for n = 1..2500</a>
%H John D. Baum, <a href="https://www.jstor.org/stable/2690056">A Number-Theoretic Sum</a>, Mathematics Magazine 55.2 (1982): 111-113.
%H P. G. Brown, <a href="http://www.jstor.org/stable/3621931">Some comments on inverse arithmetic functions</a>, Math. Gaz. 89 (516) (2005) 403-408.
%H Constantin M. Petridi, <a href="https://arxiv.org/abs/1612.07632">The Sums of the k-powers of the Euler set and their connection with Artin's conjecture for primitive roots</a>, arXiv:1612.07632 [math.NT], 2016-2018.
%F 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.
%F a(n) = n^2*A000010(n)/3 + n*A023900(n)/6, n>1. [Brown]
%F a(n) = A000010(n)/3 * (n^2 + (-1)^A001221(n)*A007947(n)/2)) for n>=2. - _Jaroslav Krizek_, Aug 24 2010
%F G.f. A(x) satisfies: A(x) = x*(1 + x)/(1 - x)^4 - Sum_{k>=2} k^2 * A(x^k). - _Ilya Gutkovskiy_, Mar 29 2020
%F Sum_{k=1..n} a(k) ~ n^4 / (2*Pi^2). - _Amiram Eldar_, Dec 03 2023
%p A053818 := proc(n)
%p local a,k;
%p a := 0 ;
%p for k from 1 to n do
%p if igcd(k,n) = 1 then
%p a := a+k^2 ;
%p end if;
%p end do:
%p a ;
%p end proc: # _R. J. Mathar_, Sep 26 2013
%t a[n_] := Plus @@ (Select[ Range@n, GCD[ #, n] == 1 &]^2); Array[a, 43] (* _Robert G. Wilson v_, Jul 01 2010 *)
%t a[1] = 1; a[n_] := Module[{f = FactorInteger[n], p, e}, p = f[[;; , 1]]; e = f[[;; , 2]]; (n^2/3) * Times @@ ((p - 1)*p^(e - 1)) + (n/6) * Times @@ (1 - p)]; Array[a, 100] (* _Amiram Eldar_, Dec 03 2023 *)
%o (PARI) a(n) = sum(k=1, n, k^2*(gcd(n,k) == 1)); \\ _Michel Marcus_, Jan 30 2016
%o (PARI) a(n) = {my(f = factor(n)); if(n == 1, 1, (n^2/3) * eulerphi(f) + (n/6) * prod(i = 1, #f~, 1 - f[i, 1]));} \\ _Amiram Eldar_, Dec 03 2023
%Y Cf. A023896, A053819, A053820.
%Y Cf. A143612. - _Gary W. Adamson_, Aug 27 2008
%Y Cf. A179871-A179880, A179882-A179887, A179890, A179891, A007645, A003627, A034934. - _Jaroslav Krizek_, Aug 01 2010
%Y Column k=2 of A308477.
%K nonn
%O 1,3
%A _N. J. A. Sloane_, Apr 07 2000
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