OFFSET
1,2
COMMENTS
Inverse Moebius transform of A158523. - Corrected by Antti Karttunen, Nov 26 2024
LINKS
Antti Karttunen, Table of n, a(n) for n = 1..20000
FORMULA
a(n) = n*lambda(n), where lambda is Liouville's function: A008836.
Totally multiplicative with a(p) = -p for prime p. [Jaroslav Krizek, Nov 01 2009]
Dirichlet g.f.: zeta(2*s-2)/zeta(s-1). Dirichlet inverse of A055615, all terms turned positive there. - R. J. Mathar, Apr 16 2011
a(n) = Sum_{d|n} lambda(d)*psi(d) = sum_{d|n} A008836(d)* A001615(d) = n/lambda(n). - Enrique Pérez Herrero, Sep 18 2012
EXAMPLE
a(6) = (-2)(-3) = +6, while a(8) = (-2)^3 = -8.
MATHEMATICA
Table[n (-1)^PrimeOmega[n], {n, 70}] (* Harvey P. Dale, Oct 05 2011 *)
PROG
(Haskell)
a061019 1 = 1
a061019 n = product $ map negate $ a027746_row n
-- Reinhard Zumkeller, Feb 08 2012
(PARI) a(n) = if( bitand(bigomega(n), 1), - n, n ); /* Joerg Arndt, Sep 19 2012 */
(Python)
from functools import reduce
from operator import ixor
from sympy import factorint
def A061019(n): return -n if reduce(ixor, factorint(n).values(), 0)&1 else n # Chai Wah Wu, Dec 20 2022
CROSSREFS
KEYWORD
easy,nice,sign,mult
AUTHOR
Marc LeBrun, Apr 13 2001
STATUS
approved