A061019 Negate primes in factorization of n.

1, -2, -3, 4, -5, 6, -7, -8, 9, 10, -11, -12, -13, 14, 15, 16, -17, -18, -19, -20, 21, 22, -23, 24, 25, 26, -27, -28, -29, -30, -31, -32, 33, 34, 35, 36, -37, 38, 39, 40, -41, -42, -43, -44, -45, 46, -47, -48, 49, -50, 51, -52, -53, 54, 55, 56, 57, 58, -59, 60, -61, 62, -63, 64, 65, -66, -67, -68, 69, -70

Offset: 1

Author

Marc LeBrun, Apr 13 2001

Keywords

• easy
• nice
• sign
• mult

Comments

Inverse Moebius transform of A158523. - Corrected by Antti Karttunen, Nov 26 2024

Examples

			a(6) = (-2)(-3) = +6, while a(8) = (-2)^3 = -8.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A000027, A001222, A061020, A001615, A158523 (Möbius transform).

Cf. A027746.

Cf. A239122 (partial sums).

Programs

Haskell

a061019 1 = 1
a061019 n = product $ map negate $ a027746_row n
-- Reinhard Zumkeller, Feb 08 2012

Mathematica

Table[n (-1)^PrimeOmega[n],{n,70}] (* Harvey P. Dale, Oct 05 2011 *)

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

Formula

a(n) = n*lambda(n), where lambda is Liouville's function: A008836.

a(n) = (-1)^(number of primes dividing n)*n = n * (-1)^A001222(n) = n*A008836(n).

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