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A065958 a(n) = n^2*Product_{distinct primes p dividing n} (1+1/p^2). 12
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

The sequence is considered to be psi_2, a generalization of Dedekind Psi Function, where psi_1 is A001615. - Enrique Pérez Herrero, Jul 06 2011

REFERENCES

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.

LINKS

E. Pérez Herrero,Table of n, a(n) for n=1..10000

FORMULA

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). - Paul D. Hanna and Vladeta Jovovic, Feb 14 2009

From Enrique Pérez 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

MAPLE

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;

MATHEMATICA

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 *)

PROG

(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 */

CROSSREFS

Cf. A000010, A001615, A007434, A065959, A065960.

Sequence in context: A072703 A086761 A045191 * A065969 A027884 A236391

Adjacent sequences:  A065955 A065956 A065957 * A065959 A065960 A065961

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane, Dec 08 2001

STATUS

approved

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Last modified August 29 02:15 EDT 2015. Contains 261184 sequences.