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%I #38 Feb 06 2022 20:38:59
%S 1,17,82,272,626,1394,2402,4352,6642,10642,14642,22304,28562,40834,
%T 51332,69632,83522,112914,130322,170272,196964,248914,279842,356864,
%U 391250,485554,538002,653344,707282,872644,923522,1114112,1200644
%N a(n) = n^4*Product_{distinct primes p dividing n} (1+1/p^4).
%H E. Pérez Herrero, <a href="/A065960/b065960.txt">Table of n, a(n) for n=1..10000</a>
%H F. A. Lewis and others, <a href="https://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^(4*e)+p^(4*e-4). - _Vladeta Jovovic_, Dec 09 2001
%F a(n) = n^4 * Sum_{d|n} mu(d)^2/d^4. - _Benoit Cloitre_, Apr 07 2002
%F a(n) = J_8(n)/J_4(n) = A069093(n)/A059377(n), where J_k is the k-th Jordan Totient Function. - _Enrique Pérez Herrero_, Aug 29 2010
%F Dirichlet g.f.: zeta(s)*zeta(s-4)/zeta(2*s). - _R. J. Mathar_, Jun 06 2011
%F From _Vaclav Kotesovec_, Sep 19 2020: (Start)
%F Sum_{k=1..n} a(k) ~ 18711*zeta(5)*n^5 / Pi^10.
%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^4/(p^8-1)) = 1.078178802583045599985995264729541574821218371712364313741065126120993131... (End)
%p A065960 := proc(n) n^4*mul(1+1/p^4,p=numtheory[factorset](n)) ; end proc:
%p seq(A065960(n),n=1..20) ; # _R. J. Mathar_, Jun 06 2011
%t a[n_] := n^4*DivisorSum[n, MoebiusMu[#]^2/#^4&]; Array[a, 40] (* _Jean-François Alcover_, Dec 01 2015 *)
%t f[p_, e_] := p^(4*e) + p^(4*(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^4*sumdiv(n,d,moebius(d)^2/d^4),","))
%Y Cf. A000010, A007434, A059377, A069093.
%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), A065959 (k=3), this sequence (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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