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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

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

F. A. Lewis and others, Problem 4002, Amer. Math. Monthly, Vol. 49, No. 9, Nov. 1942, pp. 618-619.

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

G.f.: Sum_{k>=1} mu(k)^2*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, Oct 24 2018

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, A156733, A301978, A301980.

Sequence in context: A072703 A086761 A045191 * A065969 A306775 A027884

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 February 24 09:21 EST 2020. Contains 332209 sequences. (Running on oeis4.)