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%I #63 Oct 21 2022 09:45:37
%S 1,5,10,20,26,50,50,80,90,130,122,200,170,250,260,320,290,450,362,520,
%T 500,610,530,800,650,850,810,1000,842,1300,962,1280,1220,1450,1300,
%U 1800,1370,1810,1700,2080,1682,2500,1850,2440,2340,2650,2210
%N a(n) = n^2*Product_{distinct primes p dividing n} (1+1/p^2).
%C The sequence may be considered as psi_2, a generalization of Dedekind psi function, where psi_1 is A001615. - _Enrique Pérez Herrero_, Jul 06 2011
%D József Sándor, Geometric Theorems, Diophantine Equations, and Arithmetic Functions, American Research Press, Rehoboth 2002, pp. 193.
%H E. Pérez Herrero, <a href="/A065958/b065958.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.
%F Multiplicative with a(p^e) = p^(2*e) + p^(2*e-2). - _Vladeta Jovovic_, Dec 09 2001
%F a(n) = n^2 * Sum_{d|n} mu(d)^2/d^2 - _Benoit Cloitre_, Apr 07 2002
%F a(n) = Sum_{d|n} mu(d)^2*d^2. - _Joerg Arndt_, Jul 06 2011
%F Inverse Euler transform of n*A156733(n). - _Paul D. Hanna_ and _Vladeta Jovovic_, Feb 14 2009
%F From _Enrique Pérez Herrero_, Aug 22 2010: (Start)
%F a(n) = J_4(n)/(phi(n)*psi(n)) = A059377(n)/(A001615(n)*A000010(n))
%F a(n) = J_4(n)/J_2(n) = A059377(n)/A007434(n), where J_k is the k-th Jordan totient function. (End)
%F Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2s). Dirichlet convolution of A008966 and A000290. - _R. J. Mathar_, Apr 10 2011
%F G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 + x^k)/(1 - x^k)^3. - _Ilya Gutkovskiy_, Oct 24 2018
%F Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + p^2/(p^4 - 1)) = 1.5421162831401587416523241690601522041445615542162573163112157073779258386... - _Vaclav Kotesovec_, Sep 19 2020
%F a(n) = Sum_{d|n} d*phi(d)*psi(n/d). - _Ridouane Oudra_, Jan 01 2021
%F From _Richard L. Ollerton_, May 07 2021: (Start)
%F a(n) = Sum_{k=1..n} psi(gcd(n,k))*n/gcd(n,k), where psi(n) = A001615(n).
%F a(n) = Sum_{k=1..n} psi(n/gcd(n,k))*gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
%F Sum_{k=1..n} a(k) ~ c * n^3, where c = 315*zeta(3)/Pi^6 = 0.393854... . - _Amiram Eldar_, Oct 19 2022
%p A065958 := proc(n) local i,j,k,t1,t2,t3; t1 := ifactors(n)[2]; t2 := n^2*mul((1+1/(t1[i][1])^2),i=1..nops(t1)); end;
%t JordanTotient[n_,k_:1]:=DivisorSum[n,#^k*MoebiusMu[n/# ]&]/;(n>0)&&IntegerQ[n]; A065958[n_]:=JordanTotient[n,4]/JordanTotient[n,2]; (* _Enrique Pérez Herrero_, Aug 22 2010 *)
%t f[p_, e_] := p^(2*e) + p^(2*(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*sumdiv(n,d,moebius(d)^2/d^2),","))
%o (PARI) a(n)=sumdiv(n,d,moebius(n/d)^2*d^2); /* _Joerg Arndt_, Jul 06 2011 */
%Y Cf. A000010, A007434, A156733, A301978, A301980, A321973.
%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), this sequence (k=2), A065959 (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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