A196524 a(n) = phi(n)*tau(n^2).

1, 3, 6, 10, 12, 18, 18, 28, 30, 36, 30, 60, 36, 54, 72, 72, 48, 90, 54, 120, 108, 90, 66, 168, 100, 108, 126, 180, 84, 216, 90, 176, 180, 144, 216, 300, 108, 162, 216, 336, 120, 324, 126, 300, 360, 198, 138, 432, 210, 300, 288, 360, 156, 378, 360, 504, 324, 252, 174, 720, 180, 270, 540, 416, 432, 540

Offset: 1

Author

R. J. Mathar, Oct 07 2011

Keywords

• mult
• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..1000
• J. M. Borwein, K.-K. S. Choi, On Dirichlet series for sums of squares, Ramanujan J., Vol. 7 (2003), pp. 95-127.
• Vaclav Kotesovec, Graph - the asymptotic ratio (1000000 terms).

Programs

Mathematica

Table[EulerPhi[n] DivisorSigma[0, n^2], {n, 70}] (* Alonso del Arte, Oct 07 2011 *)
f[p_, e_] := (2*e+1)*(p-1)*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

PARI

a(n)=numdiv(n^2)*eulerphi(n) \\ Charles R Greathouse IV, Dec 07 2011

Formula

Multiplicative with a(p^e) = (2e+1)*(p-1)*p^(e-1), e>0.

a(n) = A048691(n)*A000010(n).

Dirichlet g.f.: zeta^3(s-1)*product_{primes p} (1-3/p^s -1/p^(2s-2) +4/p^(2s-1) -1/p^(3s-2)) = zeta^2(s-1)*product_{primes p} (1 +p^(1-s) +p^(1-2s) -3p^(-s)).

Sum_{k=1..n} a(k) ~ c * log(n)^2 * n^2 / 4, where c = A256392 = Product_{primes p} (1 - 4/p^2 + 4/p^3 - 1/p^4) = 0.217778716619536378323007514119446813130797755... - Vaclav Kotesovec, Dec 18 2019

More precise asymptotics: Let f(s) = Product_{primes p} (1 - 3/p^s - 1/p^(2*s-2) + 4/p^(2*s-1) - 1/p^(3*s-2)), then Sum_{k=1..n} a(k) ~ n^2 * ((log(n)^2/4 + (3*gamma/2 - 1/4)*log(n) + 3*gamma^2/2 - 3*gamma/4 - 3*sg1/2 + 1/8)*f(2) + (log(n)/2 + 3*gamma/2 - 1/4)*f'(2) + f''(2)/4), where f(2) = A256392, f'(2) = f(2) * Sum_{primes p} (5*p - 3) * log(p) / (p^3 + p^2 - 3*p + 1) = 0.44156369228425957720874599661015191553108775903124..., f''(2) = f'(2)^2/f(2) + f(2) * Sum_{primes p} = p*(7*p^3 - 2*p^2 - 5*p + 4) * log(p)^2 / (p^3 + p^2 - 3*p + 1)^2 = -0.0925787956842332743072787717877016487612772912975..., gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). - Vaclav Kotesovec, Jun 18 2020