|
%I #38 Sep 18 2023 02:03:03
%S 1,-2,-3,-2,-5,6,-7,-2,-3,10,-11,6,-13,14,15,-2,-17,6,-19,10,21,22,
%T -23,6,-5,26,-3,14,-29,-30,-31,-2,33,34,35,6,-37,38,39,10,-41,-42,-43,
%U 22,15,46,-47,6,-7,10,51,26,-53,6,55,14,57,58,-59,-30,-61,62,21,-2,65,-66,-67,34,69,-70,-71,6,-73,74,15,38,77,-78,-79
%N Multiplicative with a(p^e) = -p.
%C Except for first term, row products of A142971. - _Mats Granvik_ and _Gary W. Adamson_, Jul 15 2008
%C Dirichlet inverse of A003968. - _Werner Schulte_, Oct 25 2018
%H Ivan Neretin, <a href="/A062953/b062953.txt">Table of n, a(n) for n = 1..10000</a>
%F a(n) = Sum_{d|n} mu(d)*sigma(d) = Sum_{d|n} A063441(d).
%F From _Enrique Pérez Herrero_, Aug 24 2010: (Start)
%F a(n) = Sum_{d|n} mu(d)*psi(d), where psi is A001615.
%F a(n) = rad(n)*(-1)^omega(n) = A007947(n)*(-1)^A001221(n). (End)
%F G.f.: Sum_{k>=1} mu(k)*sigma(k)*x^k/(1 - x^k). - _Ilya Gutkovskiy_, Feb 19 2017
%F a(n) = (n*invphi(n))/phi(n) = (n*A023900(n))/(A000010(n)). - _Amrit Awasthi_, Jan 30 2022
%F Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 1/p^s - 1/p^(s-1)). - _Amiram Eldar_, Sep 18 2023
%p with(numtheory): seq(coeff(series(add(mobius(k)*sigma(k)*x^k/(1-x^k),k=1..n),x,n+1), x, n), n = 1 .. 80); # _Muniru A Asiru_, Oct 26 2018
%t A062953[n_]:=DivisorSum[n,MoebiusMu[ # ]*DivisorSigma[1,# ]&] (* _Enrique Pérez Herrero_, Aug 24 2010 *)
%o (PARI) a(n) = sumdiv(n, d, moebius(d)*sigma(d)); \\ _Michel Marcus_, Feb 19 2017
%Y Apart from signs, essentially same as A007947.
%Y Cf. A001221, A001615, A003968, A063441, A142971.
%Y Cf. A000010, A023900.
%K mult,sign,easy
%O 1,2
%A _Vladeta Jovovic_, Jul 21 2001
|
