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%I #18 Nov 20 2022 01:54:06
%S 1,1,4,-3,6,4,8,-3,4,6,12,-12,14,8,24,-3,18,4,20,-18,32,12,24,-12,6,
%T 14,4,-24,30,24,32,-3,48,18,48,-12,38,20,56,-18,42,32,44,-36,24,24,48,
%U -12,8,6,72,-42,54,4,72,-24,80,30,60,-72,62,32,32,-3,84,48,68,-54,96,48,72,-12,74,38,24
%N L.g.f.: log(Product_{k>=1} (1 + mu(k)^2*x^k)) = Sum_{n>=1} a(n)*x^n/n, where mu() is the Moebius function (A008683).
%H Andrew Howroyd, <a href="/A300894/b300894.txt">Table of n, a(n) for n = 1..1000</a>
%F G.f.: Sum_{k>=1} mu(k)^2*k*x^k/(1 + x^k).
%F a(n) = n + 1 if n is an odd prime (A065091).
%F Multiplicative with a(2^e) = 1 if e = 1, and -3 otherwise, and a(p^e) = p+1 for an odd prime p. - _Amiram Eldar_, Oct 25 2020
%F Sum_{k=1..n} a(k) ~ n^2/4. - _Amiram Eldar_, Nov 20 2022
%e L.g.f.: L(x) = x + x^2/2 + 4*x^3/3 - 3*x^4/4 + 6*x^5/5 + 4*x^6/6 + 8*x^7/7 - 3*x^8/8 + 4*x^9/9 + 6*x^10/10 + ...
%e exp(L(x)) = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + 5*x^10 + ... + A087188(n)*x^n + ...
%t nmax = 75; Rest[CoefficientList[Series[Log[Product[(1 + MoebiusMu[k]^2 x^k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
%t nmax = 75; Rest[CoefficientList[Series[Sum[MoebiusMu[k]^2 k x^k/(1 + x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%t Table[DivisorSum[n, (-1)^(n/# + 1) # &, SquareFreeQ[#] &], {n, 75}]
%t f[p_, e_] := If[p == 2, If[e == 1, 1, -3], p + 1]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 25 2020 *)
%o (PARI) seq(n)=Vec(sum(k=1, n, moebius(k)^2*k*x^k/(1 + x^k) + O(x*x^n))); \\ _Andrew Howroyd_, Jul 20 2018
%o (PARI) a(n)={sumdiv(n, d, if(issquarefree(d), (-1)^(n/d + 1) * d))} \\ _Andrew Howroyd_, Jul 20 2018
%Y Cf. A005117, A008683, A048250, A065091, A087188, A300852.
%K sign,mult,easy
%O 1,3
%A _Ilya Gutkovskiy_, Mar 14 2018
%E Keyword:mult added by _Andrew Howroyd_, Jul 20 2018
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