This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A319132 #39 Jan 03 2023 09:21:46
%S A319132 1,4,5,7,7,20,9,10,9,28,13,35,15,36,35,13,19,36,21,49,45,52,25,50,13,
%T A319132 60,13,63,31,140,33,16,65,76,63,63,39,84,75,70,43,180,45,91,63,100,49,
%U A319132 65,17,52,95,105,55,52,91,90,105,124,61,245,63,132,81,19,105,260,69,133,125,252
%N A319132 a(n) = Sum_{d|n} Sum_{j|d} mu(j)^2*j, where mu = Möbius function (A008683).
%C A319132 Inverse Möbius transform of A048250.
%H A319132 Metin Sariyar, <a href="/A319132/b319132.txt">Table of n, a(n) for n = 1..16000</a>
%H A319132 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%F A319132 G.f.: Sum_{k>=1} A048250(k)*x^k/(1 - x^k).
%F A319132 L.g.f.: -log(Product_{k>=1} (1 - x^k)^(A048250(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
%F A319132 a(p^k) = (p + 1)*k + 1, where p is a prime.
%F A319132 a(n) = Sum_{d|n} mu(d)^2*d*tau(n/d). - _Ridouane Oudra_, Nov 13 2019
%F A319132 Multiplicative with a(p^e) = (p+1)*e+1. - _Amiram Eldar_, Oct 25 2020
%F A319132 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 = 0.822467... (A072691). - _Amiram Eldar_, Nov 13 2022
%F A319132 Dirichlet g.f.: zeta(s)^2*zeta(s-1)/zeta(2*s-2). - _Amiram Eldar_, Jan 03 2023
%p A319132 with(numtheory): seq(add(mobius(d)^2*d*tau(n/d), d in divisors(n)), n=1..70); # _Ridouane Oudra_, Nov 13 2019
%t A319132 Table[Sum[Sum[MoebiusMu[j]^2 j, {j, Divisors[d]}], {d, Divisors[n]}], {n, 70}]
%t A319132 nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, # &, SquareFreeQ[#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
%t A319132 nmax = 70; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(DivisorSum[k, # &, SquareFreeQ[#] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]]
%t A319132 f[p_, e_] := (p + 1)*e + 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 25 2020 *)
%o A319132 (PARI) a(n) = sumdiv(n, d, moebius(d)^2*d*numdiv(n/d)); \\ _Michel Marcus_, Nov 13 2019; corrected Jun 13 2022
%o A319132 (Magma) [&+[&+[MoebiusMu(j)^2*j:j in Divisors(d)]:d in Divisors(n)]:n in [1..70]]; // _Marius A. Burtea_, Nov 13 2019
%o A319132 (Magma) [&+[MoebiusMu(d)^2*d*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // _Marius A. Burtea_, Nov 13 2019
%Y A319132 Cf. A007429, A008683, A048250, A048691, A055615, A072691, A319131.
%K A319132 nonn,mult,easy
%O A319132 1,2
%A A319132 _Ilya Gutkovskiy_, Sep 11 2018