A327668 a(n) = n * Sum_{d|n} (-1)^(bigomega(d) - omega(d)) / d.

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, 42, 34, 40, 30, 72, 32, 43, 48, 54, 48, 55, 38, 60, 56, 66, 42, 96, 44, 60, 66, 72, 48, 84, 55, 87, 72, 70, 54, 102, 72, 88, 80, 90, 60, 120, 62, 96, 88, 85, 84, 144, 68, 90, 96, 144

Offset: 1

Author

Ilya Gutkovskiy, Sep 21 2019

Keywords

• nonn
• mult

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..20000

Crossrefs

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

Programs

Maple

with(numtheory):
a:= n-> n*add((-1)^(bigomega(d)-nops(factorset(d)))/d, d=divisors(n)):
seq(a(n), n=1..80);  # Alois P. Heinz, Sep 21 2019

Mathematica

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

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

Formula

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

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

From Amiram Eldar, Dec 01 2022: (Start)

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

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)