A057434 - OEIS

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Other ways to Give Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!) A057434 a(n) = Sum_{k=1..n} phi(k)^2. 12

%I #34 Dec 18 2019 03:22:20

%S 1,2,6,10,26,30,66,82,118,134,234,250,394,430,494,558,814,850,1174,

%T 1238,1382,1482,1966,2030,2430,2574,2898,3042,3826,3890,4790,5046,

%U 5446,5702,6278,6422,7718,8042,8618,8874,10474,10618,12382,12782

%N a(n) = Sum_{k=1..n} phi(k)^2.

%C Partial sums of A127473. - _R. J. Mathar_, Sep 29 2008

%H Ivan Neretin, <a href="/A057434/b057434.txt">Table of n, a(n) for n = 1..10000</a>

%H U. Balakrishnan & Y.-F. S. Pétermann, <a href="http://matwbn.icm.edu.pl/ksiazki/aa/aa75/aa7512.pdf">The Dirichlet series of zeta(s)*zeta(s+1)^alpha*f(s+1): On an error term associated with its coefficients</a>, Acta Arith. 75 (1996), 39--69.

%F We can derive an asymptotic formula from a general formula given in the reference, namely: a(n) = C*n^3 + O(log(x)^(4/3)log(log(x))^(8/3)) where C = (1/3)/zeta(2)^2*Product_{p prime}(1+1/(p-1)/(p+1)^2) = 0.142749835225698(...). - _Benoit Cloitre_, Dec 22 2015

%F a(n) ~ c * n^3 / 3, where c = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.4282495056770944402187657075818235461212985133559361440319... - _Vaclav Kotesovec_, Dec 18 2019

%t FoldList[Plus, 1, EulerPhi[Range[2, 50]]^2] (* _Ivan Neretin_, May 30 2015 *)

%o (PARI) a(n) = sum(k=1, n, eulerphi(k)^2); \\ _Michel Marcus_, Dec 20 2015

%Y Cf. A000010, A002088, A061502, A072379, A074789.

%K nonn

%O 1,2

%A _N. J. A. Sloane_, Sep 08 2000

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