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A065958
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n^2*Product_{distinct primes p dividing n} (1+1/p^2).
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12
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1, 5, 10, 20, 26, 50, 50, 80, 90, 130, 122, 200, 170, 250, 260, 320, 290, 450, 362, 520, 500, 610, 530, 800, 650, 850, 810, 1000, 842, 1300, 962, 1280, 1220, 1450, 1300, 1800, 1370, 1810, 1700, 2080, 1682, 2500, 1850, 2440, 2340, 2650, 2210
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OFFSET
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1,2
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COMMENTS
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The sequence is considered to be psi_2, a generalization of Dedekind Psi Function, where psi_1 is A001615. - Enrique Perez Herrero, Jul 06 2011
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REFERENCES
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F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.
József Sándor, Geometric Theorems, Diophantine Equations, and Arithmetic Functions, American Research Press, Rehoboth 2002, pp. 193.
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LINKS
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E. Pérez Herrero,Table of n, a(n) for n=1..10000
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FORMULA
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Multiplicative with a(p^e) = p^(2*e)+p^(2*e-2). - Vladeta Jovovic, Dec 09 2001
a(n) = n^2*sum(d|n, mu(d)^2/d^2) - Benoit Cloitre, Apr 07 2002
a(n) = sum(d|n, mu(d)^2*d^2). [Joerg Arndt, Jul 06 2011]
Inverse Euler transform of n*A156733(n). [From Paul D. Hanna and Vladeta Jovovic (vladeta(AT)eunet.yu), Feb 14 2009]
Contribution from Enrique Perez Herrero, Aug 22 2010: (Start)
a(n)=J_4(n)/(phi(n)*psi(n))=A059377(n)/(A001615(n)*A000010(n))
a(n)=J_4(n)/J_2(n)=A059377(n)/A007434(n), where J_k is the k-th Jordan Totient Function (End)
Dirichlet g.f. zeta(s)*zeta(s-2)/zeta(2s). Dirichlet convolution of A008966 and A000290. - R. J. Mathar, Apr 10 2011
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MAPLE
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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;
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MATHEMATICA
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JordanTotient[n_, k_:1]:=DivisorSum[n, #^k*MoebiusMu[n/# ]&]/; (n>0)&&IntegerQ[n]; A065958[n_]:=JordanTotient[n, 4]/JordanTotient[n, 2]; (* Enrique Perez Herrero, Aug 22 2010 *)
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PROG
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(PARI) for(n=1, 100, print1(n*sumdiv(n, d, moebius(d)^2/d^2), ", "))
(PARI) a(n)=sumdiv(n, d, moebius(n/d)^2*d^2); /* Joerg Arndt, Jul 06 2011 */
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CROSSREFS
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Cf. A000010, A001615, A007434, A065959, A065960.
Sequence in context: A072703 A086761 A045191 * A065969 A027884 A026357
Adjacent sequences: A065955 A065956 A065957 * A065959 A065960 A065961
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KEYWORD
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nonn,mult,changed
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AUTHOR
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N. J. A. Sloane, Dec 08 2001
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STATUS
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approved
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