A319132 - OEIS

%I #39 Jan 03 2023 09:21:46

%S 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 60,13,63,31,140,33,16,65,76,63,63,39,84,75,70,43,180,45,91,63,100,49,

%U 65,17,52,95,105,55,52,91,90,105,124,61,245,63,132,81,19,105,260,69,133,125,252

%N a(n) = Sum_{d|n} Sum_{j|d} mu(j)^2*j, where mu = Möbius function (A008683).

%C Inverse Möbius transform of A048250.

%H Metin Sariyar, <a href="/A319132/b319132.txt">Table of n, a(n) for n = 1..16000</a>

%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.

%F G.f.: Sum_{k>=1} A048250(k)*x^k/(1 - x^k).

%F L.g.f.: -log(Product_{k>=1} (1 - x^k)^(A048250(k)/k)) = Sum_{n>=1} a(n)*x^n/n.

%F a(p^k) = (p + 1)*k + 1, where p is a prime.

%F a(n) = Sum_{d|n} mu(d)^2*d*tau(n/d). - _Ridouane Oudra_, Nov 13 2019

%F Multiplicative with a(p^e) = (p+1)*e+1. - _Amiram Eldar_, Oct 25 2020

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 = 0.822467... (A072691). - _Amiram Eldar_, Nov 13 2022

%F Dirichlet g.f.: zeta(s)^2*zeta(s-1)/zeta(2*s-2). - _Amiram Eldar_, Jan 03 2023

%p with(numtheory): seq(add(mobius(d)^2*d*tau(n/d), d in divisors(n)), n=1..70); # _Ridouane Oudra_, Nov 13 2019

%t Table[Sum[Sum[MoebiusMu[j]^2 j, {j, Divisors[d]}], {d, Divisors[n]}], {n, 70}]

%t nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, # &, SquareFreeQ[#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]

%t 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 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 (PARI) a(n) = sumdiv(n, d, moebius(d)^2*d*numdiv(n/d)); \\ _Michel Marcus_, Nov 13 2019; corrected Jun 13 2022

%o (Magma) [&+[&+[MoebiusMu(j)^2*j:j in Divisors(d)]:d in Divisors(n)]:n in [1..70]]; // _Marius A. Burtea_, Nov 13 2019

%o (Magma) [&+[MoebiusMu(d)^2*d*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // _Marius A. Burtea_, Nov 13 2019

%Y Cf. A007429, A008683, A048250, A048691, A055615, A072691, A319131.

%K nonn,mult,easy

%O 1,2

%A _Ilya Gutkovskiy_, Sep 11 2018