|
|
A061019
|
|
Negate primes in factorization of n.
|
|
10
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
Inverse Moebius transform of A001615.
|
|
LINKS
|
|
|
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
|
|
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
(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
|
|
|
STATUS
|
approved
|
|
|
|