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%I #45 Feb 06 2022 20:40:57
%S 1,9,28,72,126,252,344,576,756,1134,1332,2016,2198,3096,3528,4608,
%T 4914,6804,6860,9072,9632,11988,12168,16128,15750,19782,20412,24768,
%U 24390,31752,29792,36864,37296,44226,43344,54432,50654,61740,61544
%N a(n) = n^3*Product_{distinct primes p dividing n} (1+1/p^3).
%H Enrique Pérez Herrero, <a href="/A065959/b065959.txt">Table of n, a(n) for n=1..10000</a>
%H F. A. Lewis and others, <a href="http://www.jstor.org/stable/2303350">Problem 4002</a>, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Dedekind_psi_function">Dedekind psi function</a>.
%F Multiplicative with a(p^e) = p^(3*e)+p^(3*e-3). - _Vladeta Jovovic_, Dec 09 2001
%F a(n) = n^3*Sum_{d|n} mu(d)^2/d^3. - _Benoit Cloitre_, Apr 07 2002
%F a(n) = Sum_{d|n} mu(n/d)^2*d^3. - _Joerg Arndt_, Jul 06 2011
%F a(n) = J_6(n)/J_3(n) = A069091(n)/A059376(n). - _Enrique Pérez Herrero_, Aug 22 2010
%F Dirichlet g.f.: zeta(s)*zeta(s-3)/zeta(2*s). Dirichlet convolution of A008966 and A000578. - _R. J. Mathar_, Apr 10 2011
%F G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 + 4*x^k + x^(2*k))/(1 - x^k)^4. - _Ilya Gutkovskiy_, Oct 24 2018
%F From _Vaclav Kotesovec_, Sep 19 2020: (Start)
%F Sum_{k=1..n} a(k) ~ 105*n^4 / (4*Pi^4).
%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^3/(p^6-1)) = 1.18370753651668075930203278269930233284040397061087910806697928843547863257... (End)
%t JordanTotient[n_,k_:1] := DivisorSum[n, #^k * MoebiusMu[n/#] &]/;(n>0) && IntegerQ[n]; A065959[n_] := JordanTotient[n,6] / JordanTotient[n,3]; Array[A065959, 39] (* _Enrique Pérez Herrero_, Aug 22 2010 *)
%t f[p_, e_] := p^(3*e) + p^(3*(e-1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 12 2020 *)
%o (PARI) for(n=1,100,print1(n^3*sumdiv(n,d,moebius(d)^2/d^3),","))
%o (PARI) a(n)=sumdiv(n,d,moebius(n/d)^2*d^3); \\ _Joerg Arndt_, Jul 06 2011
%Y Cf. A000010, A007434.
%Y Sequences of the form n^k * Product_ {p|n, p prime} (1 + 1/p^k) for k=0..10: A034444 (k=0), A001615 (k=1), A065958 (k=2), this sequence (k=3), A065960 (k=4), A351300 (k=5), A351301 (k=6), A351302 (k=7), A351303 (k=8), A351304 (k=9), A351305 (k=10).
%K nonn,mult,easy
%O 1,2
%A _N. J. A. Sloane_, Dec 08 2001
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