|
%I #20 Oct 25 2020 03:04:53
%S 1,-1,-4,0,-6,4,-8,0,3,6,-12,0,-14,8,24,0,-18,-3,-20,0,32,12,-24,0,5,
%T 14,0,0,-30,-24,-32,0,48,18,48,0,-38,20,56,0,-42,-32,-44,0,-18,24,-48,
%U 0,7,-5,72,0,-54,0,72,0,80,30,-60,0,-62,32,-24,0,84,-48,-68,0,96
%N a(n) = Sum_{d|n, d odd} d * mu(d) * mu(n/d).
%C Dirichlet inverse of A000593.
%H Amiram Eldar, <a href="/A327278/b327278.txt">Table of n, a(n) for n = 1..10000</a>
%F G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} A000593(k) * A(x^k).
%F Dirichlet g.f.: 1 / (zeta(s) * zeta(s-1) * (1 - 2^(1-s))).
%F a(1) = 1; a(n) = -Sum_{d|n, d<n} A000593(n/d) * a(d).
%F a(n) = Sum_{d|n} A067856(n/d) * A055615(d).
%F Multiplicative with a(2^e) = -1 if e = 1 and 0 otherwise, and a(p^e) = -(p+1) if e = 1, p if e = 2 and 0 otherwise, for an odd prime p. - _Amiram Eldar_, Oct 25 2020
%t Table[DivisorSum[n, # MoebiusMu[#] MoebiusMu[n/#] &, OddQ[#] &], {n, 1, 69}]
%t a[n_] := If[n == 1, n, -Sum[If[d < n, DivisorSum[n/d, Mod[#, 2] # &] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 69}]
%t f[p_, e_] := If[p == 2, -Boole[e == 1], Which[e == 1, -p - 1, e == 2, p, e > 2, 0]]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Oct 25 2020 *)
%o [&+[d*MoebiusMu(d)*MoebiusMu(n div d): d in [a:a in Divisors(n)| IsOdd(a)]]:n in [1..70]]; // _Marius A. Burtea_, Sep 15 2019
%Y Cf. A000593, A008683, A046692, A055615, A067856, A206787, A327276.
%K sign,mult,easy
%O 1,3
%A _Ilya Gutkovskiy_, Sep 15 2019
|
