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A053822 Dirichlet inverse of sigma_2 function (A001157). 4
1, -5, -10, 4, -26, 50, -50, 0, 9, 130, -122, -40, -170, 250, 260, 0, -290, -45, -362, -104, 500, 610, -530, 0, 25, 850, 0, -200, -842, -1300, -962, 0, 1220, 1450, 1300, 36, -1370, 1810, 1700, 0, -1682, -2500, -1850, -488, -234, 2650, -2210, 0, 49, -125, 2900, -680 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

sigma_2(n) is the sum of the squares of the divisors of n (A001157).

REFERENCES

T. M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976, page 39.

LINKS

Robert Israel, Table of n, a(n) for n = 1..10000

FORMULA

Dirichlet g.f.: 1/(zeta(x)zeta(x-2))

Multiplicative with a(p^1) = -1-p^2, a(p^2) = p^2, a(p^e) = 0 for e>=3. - Mitch Harris, Jun 27 2005

a(n) = Sum_{d|n} mu(n/d)*mu(d)*d^2. - Ilya Gutkovskiy, Nov 06 2018

MAPLE

f1:= proc(p, e) if e = 1 then -1-p^2 elif e=2 then p^2 else 0 fi end proc:

f:= n -> mul(f1(t[1], t[2]), t=ifactors(n)[2]);

map(f, [$1..100]); # Robert Israel, Jan 29 2018

MATHEMATICA

a[n_] := Sum[MoebiusMu[n/d] MoebiusMu[d] d^2, {d, Divisors[n]}];

Array[a, 100] (* Jean-Fran├žois Alcover, Mar 05 2019, after Ilya Gutkovskiy *)

PROG

(PARI) seq(n)={dirdiv(vector(n, n, n==1), vector(n, n, sigma(n, 2)))} \\ Andrew Howroyd, Aug 05 2018

CROSSREFS

Cf. A001157, A046692.

Sequence in context: A054513 A295121 A066200 * A262922 A276652 A137404

Adjacent sequences:  A053819 A053820 A053821 * A053823 A053824 A053825

KEYWORD

sign,mult,look

AUTHOR

N. J. A. Sloane, Apr 08 2000

STATUS

approved

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Last modified October 23 12:19 EDT 2019. Contains 328345 sequences. (Running on oeis4.)