A055615 a(n) = n * mu(n), where mu is the Möbius function A008683.

1, -2, -3, 0, -5, 6, -7, 0, 0, 10, -11, 0, -13, 14, 15, 0, -17, 0, -19, 0, 21, 22, -23, 0, 0, 26, 0, 0, -29, -30, -31, 0, 33, 34, 35, 0, -37, 38, 39, 0, -41, -42, -43, 0, 0, 46, -47, 0, 0, 0, 51, 0, -53, 0, 55, 0, 57, 58, -59, 0, -61, 62, 0, 0, 65, -66, -67, 0, 69, -70, -71, 0

Offset

1,2

Comments

Dirichlet inverse of n (A000027).

Absolute values give n if n is squarefree, otherwise 0.

a(n) is multiplicative because both mu(n) and n are. - Mitch Harris, Jun 09 2005

a(n) is multiplicative with a(p^1) = -p, a(p^e) = 0 if e > 1. - David W. Wilson, Jun 12 2005

Negative of the Moebius number of the dihedral group of order 2n. - Eric M. Schmidt, Jul 28 2013

Links

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

Mats Granvik, Primes approximated by eigenvalues.

Mats Granvik, Mobius function times n approximated by eigenvalues.

Formula

a(n) = n * A008683(n).

Dirichlet g.f.: 1/zeta(s-1).

Multiplicative with a(p^e) = -p*0^(e-1), e>0 and p prime. - Reinhard Zumkeller, Jul 17 2003

Conjectures: lim b->1+ Sum n=1..inf a(n)*b^(-n) = -12 and lim b->1- Sum n=1..inf a(n)*b^n = -12 (+ indicates that b decreases to 1, - indicates it increases to 1), both considering that zeta(-1) = -1/12 and calculations (more generally mu(n)*n^s is Abel summable to zeta(-s)). - Gerald McGarvey, Sep 26 2004

Dirichlet generating function for the absolute value: zeta(s-1)/zeta(2s-2). - Franklin T. Adams-Watters, Sep 11 2005

G.f. A(x) satisfies: A(x) = x - Sum_{k>=2} k*A(x^k). - Ilya Gutkovskiy, May 11 2019

Sum_{k=1..n} abs(a(k)) ~ 3*n^2/Pi^2. - Amiram Eldar, Feb 02 2024

Example

G.f. = x - 2*x^2 - 3*x^3 - 5*x^5 + 6*x^6 - 7*x^7 + 10*x^10 - 11*x^11 - 13*x^13 + ...

Maple

with(numtheory): A055615:=n->n*mobius(n): seq(A055615(n), n=1..100); # Wesley Ivan Hurt, Nov 18 2014

Mathematica

Table[n MoebiusMu[n], {n, 80}] (* Harvey P. Dale, May 26 2011 *)

Prog

(PARI) {a(n) = if( n<1, 0, n * moebius(n))};
(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 - p*X)[n])};
(Magma) [n*MoebiusMu(n): n in [1..80]]; // Vincenzo Librandi, Nov 19 2014
(Haskell)
a055615 n = a008683 n * n  -- Reinhard Zumkeller, Sep 04 2015
(SageMath) [n*moebius(n) for n in (1..100)] # G. C. Greubel, May 24 2022
(Python)
from sympy import mobius
def A055615(n): return n*mobius(n) # Chai Wah Wu, Apr 01 2023

Crossrefs

Moebius transform of A023900.

Cf. A000027 (Dirichlet inverse), A061669 (sum with it).

Cf. A062004.

Cf. A013929 (positions of 0's), A068340 (partial sums), A261869 (first differences), A261890 (second differences).

Cf. A008683, A334657, A334659, A334660.

Sequence in context: A248092 A145105 A140700 * A378655 A366390 A243059

Adjacent sequences: A055612 A055613 A055614 * A055616 A055617 A055618

Keyword

sign,easy,nice,mult

Author

Michael Somos, Jun 04 2000

Status

approved