|
%I #44 Sep 21 2020 02:46:06
%S 1,3,6,10,12,18,18,28,30,36,30,60,36,54,72,72,48,90,54,120,108,90,66,
%T 168,100,108,126,180,84,216,90,176,180,144,216,300,108,162,216,336,
%U 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
%N a(n) = phi(n)*tau(n^2).
%H Vincenzo Librandi, <a href="/A196524/b196524.txt">Table of n, a(n) for n = 1..1000</a>
%H J. M. Borwein, K.-K. S. Choi, <a href="http://dx.doi.org/10.1023/A:1026230709945">On Dirichlet series for sums of squares</a>, Ramanujan J., Vol. 7 (2003), pp. 95-127.
%H Vaclav Kotesovec, <a href="/A196524/a196524.jpg">Graph - the asymptotic ratio (1000000 terms)</a>.
%F Multiplicative with a(p^e) = (2e+1)*(p-1)*p^(e-1), e>0.
%F a(n) = A048691(n)*A000010(n).
%F 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)).
%F 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
%F 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
%t Table[EulerPhi[n] DivisorSigma[0, n^2], {n, 70}] (* _Alonso del Arte_, Oct 07 2011 *)
%t 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 *)
%o (PARI) a(n)=numdiv(n^2)*eulerphi(n) \\ _Charles R Greathouse IV_, Dec 07 2011
%K mult,nonn,easy
%O 1,2
%A _R. J. Mathar_, Oct 07 2011
|
