A226177 -id:A226177 - cp's OEIS frontend

A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).

1, -1, -1, -1, -1, 1, -1, -1, -1, 1, -1, 1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -1, 1, -1, -1, -1, -1, 1, 1, 1, 1, -1, 1, 1, 1, -1, -1, -1, 1, 1, 1, -1, 1, -1, 1, 1, 1, -1, 1, 1, 1, 1, 1, -1, -1, -1, 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, 1, 1, -1, -1, 1, -1, 1, -1, -1, 1, 1, 1

Offset: 1

Reinhard Zumkeller, Oct 14 2002

Multiplicative: a(1) = 1, a(n) for n >=2 is sign of parity of number of distinct primes dividing n. a(p) = -1, a(pq) = 1, a(pq...z) = (-1)^k, a(p^k) = -1, where p,q,.. z are distinct primes and k natural numbers. - Jaroslav Krizek, Mar 17 2009

a(n) is the unitary Moebius function, i.e., the inverse of the constant 1 function under the unitary convolution defined by (f X g)(n)= sum of f(d)g(n/d), where the sum is over the unitary divisors d of n (divisors d of n such that gcd(d,n/d)=1). - Laszlo Toth, Oct 08 2009

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eckford Cohen, Arithmetical functions associated with the unitary divisors of an integer, Math. Zeitschr., Vol. 74 (1960), pp. 66-80.
Jan van de Lune and Robert E. Dressler, Some theorems concerning the number theoretic function omega(n), Journal für die reine und angewandte Mathematik, Vol. 277 (1975), pp. 117-119; alternative link.

Cf. A000005, A000010, A001221, A007947, A008683, A008836, A030230, A065469, A076480, A180403, A226177.

Cf. A174863 (partial sums).

a076479 = a008683 . a007947  -- Reinhard Zumkeller, Jun 01 2013

[(-1)^(#PrimeDivisors(n)): n in [1..100]]; // Vincenzo Librandi, Dec 31 2018

A076479 := proc(n)
    (-1)^A001221(n) ;
end proc:
seq(A076479(n),n=1..80) ; # R. J. Mathar, Nov 02 2016

Table[(-1)^PrimeNu[n], {n,50}] (* Enrique Pérez Herrero, Jan 17 2013 *)

N=66;
mu=vector(N); mu[1]=1;
{ for (n=2,N,
    s = 0;
    fordiv (n,d,
        if (gcd(d,n/d)!=1, next() ); /* unitary divisors only */
        s += mu[d];
    );
    mu[n] = -s;
); };
mu /* Joerg Arndt, May 13 2011 */
/* omitting the line if ( gcd(...)) gives the usual Moebius function */

a(n)=(-1)^omega(n) \\ Charles R Greathouse IV, Aug 02 2013

from math import prod
from sympy.ntheory import mobius, primefactors
def A076479(n): return mobius(prod(primefactors(n))) # Chai Wah Wu, Sep 24 2021

a(n) = A008683(A007947(n)).
a(n) = (-1)^A001221(n). Multiplicative with a(p^e) = -1. - Vladeta Jovovic, Dec 03 2002
a(n) = sign(A180403(n)). - Mats Granvik, Oct 08 2010
Sum_{n>=1} a(n)*phi(n)/n^3 = A065463 with phi()=A000010() [Cohen, Lemma 3.5]. - R. J. Mathar, Apr 11 2011
Dirichlet convolution of A000012 with A226177 (signed variant of A074823 with one factor mu(n) removed). - R. J. Mathar, Apr 19 2011
Sum_{n>=1} a(n)/n^2 = A065469. - R. J. Mathar, Apr 19 2011
a(n) = Sum_{d|n} mu(d)*tau_2(d) = Sum_{d|n} A008683(d)*A000005(d) . - Enrique Pérez Herrero, Jan 17 2013
a(A030230(n)) = -1; a(A030231(n)) = +1. - Reinhard Zumkeller, Jun 01 2013
Dirichlet g.f.: zeta(s) * Product_{p prime} (1 - 2p^(-s)). - Álvar Ibeas, Dec 30 2018
Sum_{n>=1} a(n)/n = 0 (van de Lune and Dressler, 1975). - Amiram Eldar, Mar 05 2021
From Richard L. Ollerton, May 07 2021: (Start)
For n>1, Sum_{k=1..n} a(gcd(n,k))*phi(gcd(n,k))*rad(gcd(n,k))/gcd(n,k) = 0.
For n>1, Sum_{k=1..n} a(n/gcd(n,k))*phi(gcd(n,k))*rad(n/gcd(n,k))*gcd(n,k) = 0. (End)
a(n) = Sum_{d1|n} Sum_{d2|n} mu(d1*d2). - Ridouane Oudra, May 25 2023

A074823 a(n) = 2^omega(n)*mu(n)^2.

Original entry on oeis.org

1, 2, 2, 0, 2, 4, 2, 0, 0, 4, 2, 0, 2, 4, 4, 0, 2, 0, 2, 0, 4, 4, 2, 0, 0, 4, 0, 0, 2, 8, 2, 0, 4, 4, 4, 0, 2, 4, 4, 0, 2, 8, 2, 0, 0, 4, 2, 0, 0, 0, 4, 0, 2, 0, 4, 0, 4, 4, 2, 0, 2, 4, 0, 0, 4, 8, 2, 0, 4, 8, 2, 0, 2, 4, 0, 0, 4, 8, 2, 0, 0, 4, 2, 0, 4, 4, 4, 0, 2, 0, 4, 0, 4, 4, 4, 0, 2, 0, 0, 0, 2, 8, 2, 0, 8

Offset: 1

Benoit Cloitre, Sep 08 2002

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences computed from exponents in factorization of n.

Cf. A001221, A008683, A008966, A034444, A069201, A226177, A347149.

Table[MoebiusMu[n]^2 * 2^PrimeNu[n], {n, 1, 100}] (* Vaclav Kotesovec, Aug 20 2021 *)
f[p_, e_] :=If[e==1, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)

a(n) = 2^omega(n)*moebius(n)^2; \\ Michel Marcus, Jul 23 2017

for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X))[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

for(n=1, 100, print1(direuler(p=2, n, (1 - 3*X^2 + 2*X^3)/(1 - X)^2)[n], ", ")) \\ Vaclav Kotesovec, Aug 20 2021

(define (A074823 n) (if (= 1 n) n (* (if (= 1 (A067029 n)) 2 0) (A074823 (A028234 n))))) ;; Antti Karttunen, Jul 23 2017

Sum_{k=1..n} a(k) = A069201(n).
Multiplicative with a(p)=2, a(p^e)=0, e > 1.
a(n) = A034444(n)*A008966(n). - R. J. Mathar, Apr 15 2011
Sum_{n>0} a(n)/n^s = Product_{p prime} (1 + 2p^(-s)). - Ralf Stephan, Jul 07 2013
a(n) = abs(A226177(n)). - Antti Karttunen, Jul 23 2017
Dirichlet g.f.: zeta(s)^2 * Product_{primes p} (1 - 3/p^(2*s) + 2/p^(3*s)). - Vaclav Kotesovec, Aug 20 2021

Additional comments from Vladeta Jovovic, Dec 30 2002

A326814 Dirichlet g.f.: (1/zeta(s)) * Product_{p prime} (1 - 2 * p^(-s)).

Original entry on oeis.org

1, -3, -3, 2, -3, 9, -3, 0, 2, 9, -3, -6, -3, 9, 9, 0, -3, -6, -3, -6, 9, 9, -3, 0, 2, 9, 0, -6, -3, -27, -3, 0, 9, 9, 9, 4, -3, 9, 9, 0, -3, -27, -3, -6, -6, 9, -3, 0, 2, -6, 9, -6, -3, 0, 9, 0, 9, 9, -3, 18, -3, 9, -6, 0, 9, -27, -3, -6, 9, -27, -3, 0, -3, 9, -6

Offset: 1

Ilya Gutkovskiy, Oct 19 2019

Moebius transform applied twice to A076479 (unitary Moebius function).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Ilya Gutkovskiy, Scatter plot of partial sums of A326814 up to n=10000.

Cf. A001221, A007428, A008683, A046099 (positions of 0's), A076479, A182139 (Dirichlet inverse), A226177, A326415, A326815.

Table[Sum[MoebiusMu[n/d] MoebiusMu[d] 2^PrimeNu[d], {d, Divisors[n]}], {n, 1, 75}]
f[p_, e_] := Which[e == 1, -3, e == 2, 2, e > 2, 0]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

a(n) = sumdiv(n, d, moebius(n/d)*moebius(d)*2^omega(d)); \\ Michel Marcus, Oct 26 2020

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X)*(1 - X))[n], ", ")) \\ Vaclav Kotesovec, Aug 22 2021

a(n) = Sum_{d|n} mu(n/d) * mu(d) * 2^omega(d), where mu = A008683 and omega = A001221.

Multiplicative with a(p^e) = -3 if e = 1, 2 if e = 2, and 0 otherwise. - Amiram Eldar, Oct 26 2020

A062563 a(n) = Sum_{k=1..n} d(k)* mu(k), where d(k) is the number of divisors function.

Original entry on oeis.org

1, -1, -3, -3, -5, -1, -3, -3, -3, 1, -1, -1, -3, 1, 5, 5, 3, 3, 1, 1, 5, 9, 7, 7, 7, 11, 11, 11, 9, 1, -1, -1, 3, 7, 11, 11, 9, 13, 17, 17, 15, 7, 5, 5, 5, 9, 7, 7, 7, 7, 11, 11, 9, 9, 13, 13, 17, 21, 19, 19, 17, 21, 21, 21, 25, 17, 15, 15, 19, 11, 9, 9, 7, 11, 11, 11, 15, 7, 5, 5, 5, 9, 7, 7, 11, 15, 19, 19, 17, 17, 21, 21, 25, 29, 33, 33, 31

Offset: 1

Jason Earls, Jul 03 2001

sign

Partial sums of A226177.

Accumulate[Table[DivisorSigma[0,n]MoebiusMu[n],{n,100}]] (* Harvey P. Dale, Aug 15 2016 *)

v=[]; for(n=1,250,v=concat(v,sum(k=1,n,numdiv(k)*moebius(k)))); v

Also a(n) = Sum_{k=1..n} 2^omega(k)*mu(k). - Benoit Cloitre, Jun 13 2007

cp's OEIS Frontend

A076479 a(n) = mu(rad(n)), where mu is the Moebius-function (A008683) and rad is the radical or squarefree kernel (A007947).

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A074823 a(n) = 2^omega(n)*mu(n)^2.

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A326814 Dirichlet g.f.: (1/zeta(s)) * Product_{p prime} (1 - 2 * p^(-s)).

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A062563 a(n) = Sum_{k=1..n} d(k)* mu(k), where d(k) is the number of divisors function.

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A349925 Dirichlet g.f.: Product_{k>=2} (1 - 2 * k^(-s)).

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