A306408 a(n) = Sum_{d|n} (-1)^omega(n/d) * d.

1, 1, 2, 1, 4, 2, 6, 1, 5, 4, 10, 2, 12, 6, 8, 1, 16, 5, 18, 4, 12, 10, 22, 2, 19, 12, 14, 6, 28, 8, 30, 1, 20, 16, 24, 5, 36, 18, 24, 4, 40, 12, 42, 10, 20, 22, 46, 2, 41, 19, 32, 12, 52, 14, 40, 6, 36, 28, 58, 8, 60, 30, 30, 1, 48, 20, 66, 16, 44, 24, 70, 5, 72

Offset

1,3

Comments

If n is squarefree (A005117), then a(n) = A000010(n) where A000010 is the Euler totient function.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = n * Sum_{d|n} A076479(d)/d.

Sum_{k=1..n} a(k) ~ A065469 * n*(n+1)/2.

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

a(n) = Sum_{d|n} mu(d)*tau(d)*sigma(n/d). - Ridouane Oudra, Nov 24 2019

a(n) = Sum_{d|n} A058026(d). - Amiram Eldar, May 01 2020

Maple

with(numtheory): omega := n -> nops(factorset(n)):
seq(add((-1)^omega(n/d)*d, d in divisors(n)), n=1..100); # Ridouane Oudra, Nov 24 2019

Mathematica

f[p_, e_] := p^e - (p^e - 1)/(p - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 01 2020 *)

Prog

(PARI) a(n) = sumdiv(n, d, (-1)^omega(n/d) * d);
(PARI) a(n) = my(f=factor(n)); prod(k=1, #f~, f[k, 1]^f[k, 2] - (f[k, 1]^f[k, 2] - 1)/(f[k, 1]-1));

Crossrefs

Cf. A058026, A065469, A076479, A206369.

Sequence in context: A326069 A326068 A318878 * A232626 A322250 A175542

Adjacent sequences: A306405 A306406 A306407 * A306409 A306410 A306411

Keyword

nonn,mult

Author

Daniel Suteu, Apr 05 2019

Status

approved