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%I #26 Sep 19 2020 12:49:05
%S 1,6,12,12,30,72,56,24,36,180,132,144,182,336,360,48,306,216,380,360,
%T 672,792,552,288,150,1092,108,672,870,2160,992,96,1584,1836,1680,432,
%U 1406,2280,2184,720,1722,4032,1892,1584,1080,3312,2256,576,392,900
%N a(n) = n multiplied by the sum of its squarefree divisors (A048250(n)).
%C Sum of reciprocals converges to Pi^2/6. The natural density of positive integers m such that A003557(m) = n equals 6/(a(n)*Pi^2).
%C If m is coprime to 6, a(3m) = a(4m).
%C Apparently the absolute values of the Dirichlet inverse of A000082. - _R. J. Mathar_, Mar 14 2011
%H Amiram Eldar, <a href="/A181797/b181797.txt">Table of n, a(n) for n = 1..10000</a>
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Natural_density">Natural density</a>
%F a(n) = n*A048250(n). Multiplicative with a(p^e) = (p+1)*p^e.
%F Dirichlet g.f. zeta(s-1)*zeta(s-2)/zeta(2*s-4). - _R. J. Mathar_, Mar 14 2011
%F G.f.: x*f'(x), where f(x) = Sum_{k>=1} mu(k)^2*k*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Apr 10 2017
%F Sum_{k=1..n} a(k) ~ n^3 / 3. - _Vaclav Kotesovec_, Feb 02 2019
%F Sum_{k>=1} 1/a(k) = Pi^2/6. - _Vaclav Kotesovec_, Sep 19 2020
%p A181797 := proc(n) local f; f := ifactors(n)[2] ; mul( op(1,d)^op(2,d)*( op(1,d)+1),d=f) ; end proc: # _R. J. Mathar_, Dec 05 2010
%t Table[n*Sum[d*MoebiusMu[d]^2, {d, Divisors[n]}], {n, 1, 50}] (* _Vaclav Kotesovec_, Feb 02 2019 *)
%o (Sage) A181797 = lambda n: n * sum(d for d in divisors(n) if is_squarefree(d)) # _D. S. McNeil_, Dec 05 2010
%o (PARI) a(n)=n*sumdiv(n,d,d*moebius(d)^2)
%Y Cf. A181798, A181799.
%K nonn,easy,mult
%O 1,2
%A _Matthew Vandermast_, Dec 05 2010
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