A057434 a(n) = Sum_{k=1..n} phi(k)^2.

1, 2, 6, 10, 26, 30, 66, 82, 118, 134, 234, 250, 394, 430, 494, 558, 814, 850, 1174, 1238, 1382, 1482, 1966, 2030, 2430, 2574, 2898, 3042, 3826, 3890, 4790, 5046, 5446, 5702, 6278, 6422, 7718, 8042, 8618, 8874, 10474, 10618, 12382, 12782

Offset: 1

Author

N. J. A. Sloane, Sep 08 2000

Keywords

• nonn

Comments

Partial sums of A127473. - R. J. Mathar, Sep 29 2008

Links

• Ivan Neretin, Table of n, a(n) for n = 1..10000
• U. Balakrishnan & Y.-F. S. Pétermann, The Dirichlet series of zeta(s)*zeta(s+1)^alpha*f(s+1): On an error term associated with its coefficients, Acta Arith. 75 (1996), 39--69.

Crossrefs

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

Programs

Mathematica

FoldList[Plus, 1, EulerPhi[Range[2, 50]]^2] (* Ivan Neretin, May 30 2015 *)

PARI

a(n) = sum(k=1, n, eulerphi(k)^2); \\ Michel Marcus, Dec 20 2015

Formula

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

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