A327668 - OEIS

%I #11 Dec 01 2022 10:53:27

%S 1,3,4,5,6,12,8,11,11,18,12,20,14,24,24,21,18,33,20,30,32,36,24,44,29,

%T 42,34,40,30,72,32,43,48,54,48,55,38,60,56,66,42,96,44,60,66,72,48,84,

%U 55,87,72,70,54,102,72,88,80,90,60,120,62,96,88,85,84,144,68,90,96,144

%N a(n) = n * Sum_{d|n} (-1)^(bigomega(d) - omega(d)) / d.

%H Alois P. Heinz, <a href="/A327668/b327668.txt">Table of n, a(n) for n = 1..20000</a>

%F G.f.: Sum_{k>=1} mu(rad(k)) * lambda(k) * x^k / (1 - x^k)^2.

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

%F From _Amiram Eldar_, Dec 01 2022: (Start)

%F Multiplicative with a(p^e) = p^e + (p^e-(-1)^e)/(p+1).

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} ((p^2+2)/(p^2+1)) = 0.7207673679... . (End)

%p with(numtheory):

%p a:= n-> n*add((-1)^(bigomega(d)-nops(factorset(d)))/d, d=divisors(n)):

%p seq(a(n), n=1..80);  # _Alois P. Heinz_, Sep 21 2019

%t a[n_] := n Sum[(-1)^(PrimeOmega[d] - PrimeNu[d])/d, {d, Divisors[n]}]; Table[a[n], {n, 1, 70}]

%t f[p_, e_] := p^e + (p^e-(-1)^e)/(p+1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 01 2022 *)

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, f[i,1]^f[i,2] + (f[i,1]^f[i,2] - (-1)^f[i,2])/(f[i,1]+1));} \\ _Amiram Eldar_, Dec 01 2022

%Y Cf. A001221, A001222, A008836, A046660, A055076, A076479, A162511, A206369, A306408.

%K nonn,mult

%O 1,2

%A _Ilya Gutkovskiy_, Sep 21 2019