A074722 -id:A074722 - cp's OEIS frontend

A158522 Dirichlet inverse of number of unitary divisors of n (A034444).

1, -2, -2, 2, -2, 4, -2, -2, 2, 4, -2, -4, -2, 4, 4, 2, -2, -4, -2, -4, 4, 4, -2, 4, 2, 4, -2, -4, -2, -8, -2, -2, 4, 4, 4, 4, -2, 4, 4, 4, -2, -8, -2, -4, -4, 4, -2, -4, 2, -4, 4, -4, -2, 4, 4, 4, 4, 4, -2, 8, -2, 4, -4, 2, 4, -8, -2, -4, 4, -8, -2, -4, -2, 4, -4, -4, 4, -8, -2, -4, 2, 4, -2

Offset: 1

Jaroslav Krizek, Mar 20 2009

Abs{a(n)} = A034444(n). Examples of Dirichlet convolutions with function a(n), i.e., b(n) = Sum_{d|n} a(d)*c(n/d): a(n) * A034444(n) = A063524(n), a(n) * A000005(n) = A010052(n), a(n) * A000027(n) = A074722(n), a(n) * A000012(n) = A008836(n).

Möbius transform of Liouville's lambda function (A008836). - Wesley Ivan Hurt, Jun 22 2024

			a(60) = a(2^2*3*5) = [(-1)^2*2]*[(-1)^1*2]*[(-1)^1*2] = 2*(-2)*(-2) = 8.

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A034444, A063524, A000005, A010052, A000027, A074722, A001222, A001221, A000040, A006881, A120944, A000961.

Table[LiouvilleLambda[n] 2^PrimeNu[n], {n, 1, 50}] (* Geoffrey Critzer, Mar 07 2015 *)

for(n=1,20, print1((-1)^bigomega(n)* 2^omega(n), ", ")) \\ G. C. Greubel, May 21 2017

a(n) = (-1)^A001222(n)*A034444(n) = (-1)^A001222(n)*2^A001221(n), for n >= 2.
Multiplicative with a(p^e) = 2*(-1)^e, p prime, e>0. a(p^0) = 1.
Dirichlet g.f.: zeta(2s)/(zeta(s))^2. - R. J. Mathar, Apr 02 2011
a(n) = Sum_{d|n} (-1)^Omega(d) * mu(n/d). - Wesley Ivan Hurt, Jun 22 2024

A332845 a(n) = (-1)^omega(n) * Sum_{k=1..n} (-1)^omega(n/gcd(n, k)), where omega = A001221.

Original entry on oeis.org

1, 0, 1, 2, 3, 0, 5, 6, 7, 0, 9, 2, 11, 0, 3, 14, 15, 0, 17, 6, 5, 0, 21, 6, 23, 0, 25, 10, 27, 0, 29, 30, 9, 0, 15, 14, 35, 0, 11, 18, 39, 0, 41, 18, 21, 0, 45, 14, 47, 0, 15, 22, 51, 0, 27, 30, 17, 0, 57, 6, 59, 0, 35, 62, 33, 0, 65, 30, 21, 0, 69, 42, 71

Offset: 1

Ilya Gutkovskiy, Feb 26 2020

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

Cf. A000010, A001221, A016825 (positions of 0's), A049060, A058026, A074722, A076479, A307868.

Table[(-1)^PrimeNu[n] Sum[(-1)^PrimeNu[n/GCD[n, k]], {k, 1, n}], {n, 1, 73}]
Table[(-1)^PrimeNu[n] Sum[(-1)^PrimeNu[d] EulerPhi[d], {d, Divisors[n]}], {n, 1, 73}]
f[p_, e_] := p^e - 2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; s = Array[a, 100] (* Amiram Eldar, Nov 01 2022 *)

a(n) = (-1)^omega(n) * sum(k=1, n, (-1)^omega(n/gcd(n, k))); \\ Michel Marcus, Feb 26 2020

a(n) = {my(f = factor(n)); prod(i=1, #f~, f[i,1]^f[i,2] - 2); } \\ Amiram Eldar, Nov 01 2022

a(n) = (-1)^omega(n) * Sum_{d|n} (-1)^omega(d) * phi(d).
a(p) = p - 2, where p is prime.
From Amiram Eldar, Nov 01 2022: (Start)
Multiplicative with a(p^e) = p^e - 2.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/2) * Product_{p prime} (1 - 2/(p*(p+1))) = A307868 / 2 = 0.2358403068... . (End)

A333569 a(n) = Sum_{d|n} (-1)^(bigomega(d) - omega(d)) * phi(n/d).

Original entry on oeis.org

1, 2, 3, 2, 5, 6, 7, 6, 7, 10, 11, 6, 13, 14, 15, 10, 17, 14, 19, 10, 21, 22, 23, 18, 23, 26, 23, 14, 29, 30, 31, 22, 33, 34, 35, 14, 37, 38, 39, 30, 41, 42, 43, 22, 35, 46, 47, 30, 47, 46, 51, 26, 53, 46, 55, 42, 57, 58, 59, 30, 61, 62, 49, 42, 65, 66, 67, 34, 69, 70, 71, 42, 73, 74, 69

Offset: 1

Ilya Gutkovskiy, Mar 26 2020

Moebius transform of A327668.

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

Cf. A000010, A001221, A001222, A005117 (fixed points), A046660, A058026, A074722, A162511, A327666, A327668.

Table[Sum[(-1)^(PrimeOmega[d] - PrimeNu[d]) EulerPhi[n/d], {d, Divisors[n]}], {n, 1, 75}]
Table[Sum[(-1)^(PrimeOmega[GCD[n, k]] - PrimeNu[GCD[n, k]]), {k, 1, n}], {n, 1, 75}]
f[p_, e_] := If[e > 1, (p^e*(p^2+p-2) - 2*(-1)^e*p)/(p*(p + 1)), p]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a,100] (* Amiram Eldar, Nov 12 2022 *)

a(n) = sumdiv(n, d, (-1)^(bigomega(d) - omega(d)) * eulerphi(n/d)); \\ Michel Marcus, Mar 27 2020

a(n) = Sum_{k=1..n} (-1)^(bigomega(gcd(n,k)) - omega(gcd(n,k))).
a(n) = Sum_{d|n} mu(n/d) * A327668(d).
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p) = p, and a(p^e) = (p^e*(p^2+p-2) - 2*(-1)^e*p)/(p*(p+1)) for e>1.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (1/5) * Product_{p prime} (1 + 2/p^2) = 0.4381740171... . (End)

cp's OEIS Frontend

A158522 Dirichlet inverse of number of unitary divisors of n (A034444).

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A332845 a(n) = (-1)^omega(n) * Sum_{k=1..n} (-1)^omega(n/gcd(n, k)), where omega = A001221.

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A333569 a(n) = Sum_{d|n} (-1)^(bigomega(d) - omega(d)) * phi(n/d).

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