A319132 a(n) = Sum_{d|n} Sum_{j|d} mu(j)^2*j, where mu = Möbius function (A008683).
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, 60, 13, 63, 31, 140, 33, 16, 65, 76, 63, 63, 39, 84, 75, 70, 43, 180, 45, 91, 63, 100, 49, 65, 17, 52, 95, 105, 55, 52, 91, 90, 105, 124, 61, 245, 63, 132, 81, 19, 105, 260, 69, 133, 125, 252
Offset: 1
Links
- Metin Sariyar, Table of n, a(n) for n = 1..16000
- N. J. A. Sloane, Transforms.
Programs
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Magma
[&+[&+[MoebiusMu(j)^2*j:j in Divisors(d)]:d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019
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Magma
[&+[MoebiusMu(d)^2*d*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019
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Maple
with(numtheory): seq(add(mobius(d)^2*d*tau(n/d), d in divisors(n)), n=1..70); # Ridouane Oudra, Nov 13 2019
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Mathematica
Table[Sum[Sum[MoebiusMu[j]^2 j, {j, Divisors[d]}], {d, Divisors[n]}], {n, 70}] nmax = 70; Rest[CoefficientList[Series[Sum[DivisorSum[k, # &, SquareFreeQ[#] &] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]] nmax = 70; Rest[CoefficientList[Series[-Log[Product[(1 - x^k)^(DivisorSum[k, # &, SquareFreeQ[#] &]/k), {k, 1, nmax}]], {x, 0, nmax}], x] Range[0, nmax]] f[p_, e_] := (p + 1)*e + 1; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 25 2020 *) -
PARI
a(n) = sumdiv(n, d, moebius(d)^2*d*numdiv(n/d)); \\ Michel Marcus, Nov 13 2019; corrected Jun 13 2022
Formula
G.f.: Sum_{k>=1} A048250(k)*x^k/(1 - x^k).
L.g.f.: -log(Product_{k>=1} (1 - x^k)^(A048250(k)/k)) = Sum_{n>=1} a(n)*x^n/n.
a(p^k) = (p + 1)*k + 1, where p is a prime.
a(n) = Sum_{d|n} mu(d)^2*d*tau(n/d). - Ridouane Oudra, Nov 13 2019
Multiplicative with a(p^e) = (p+1)*e+1. - Amiram Eldar, Oct 25 2020
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/12 = 0.822467... (A072691). - Amiram Eldar, Nov 13 2022
Dirichlet g.f.: zeta(s)^2*zeta(s-1)/zeta(2*s-2). - Amiram Eldar, Jan 03 2023
Comments