OFFSET
1,2
COMMENTS
Inverse Möbius transform of A048250.
LINKS
Metin Sariyar, Table of n, a(n) for n = 1..16000
N. J. A. Sloane, Transforms.
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
MAPLE
with(numtheory): seq(add(mobius(d)^2*d*tau(n/d), d in divisors(n)), n=1..70); # Ridouane Oudra, Nov 13 2019
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 *)
PROG
(PARI) a(n) = sumdiv(n, d, moebius(d)^2*d*numdiv(n/d)); \\ Michel Marcus, Nov 13 2019; corrected Jun 13 2022
(Magma) [&+[&+[MoebiusMu(j)^2*j:j in Divisors(d)]:d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019
(Magma) [&+[MoebiusMu(d)^2*d*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 13 2019
CROSSREFS
KEYWORD
nonn,mult,easy
AUTHOR
Ilya Gutkovskiy, Sep 11 2018
STATUS
approved