A392285 a(n) = omega(n) * (-1)^(bigomega(n) + 1).

0, 1, 1, -1, 1, -2, 1, 1, -1, -2, 1, 2, 1, -2, -2, -1, 1, 2, 1, 2, -2, -2, 1, -2, -1, -2, 1, 2, 1, 3, 1, 1, -2, -2, -2, -2, 1, -2, -2, -2, 1, 3, 1, 2, 2, -2, 1, 2, -1, 2, -2, 2, 1, -2, -2, -2, -2, -2, 1, -3, 1, -2, 2, -1, -2, 3, 1, 2, -2, 3

Offset

1,6

Comments

a(n) is the number of distinct primes dividing n multiplied by the opposite of the Liouville's function lambda.

Möbius transform of A358769.

a(n) = 1 iff n is in A246551.

a(n) = -1 iff n is in A056798 minus {1}.

a(n) = 2 iff n is in A187039 minus {1}.

a(n) = -2 iff n is in A279458 minus {1}.

Links

Ridouane Oudra, Table of n, a(n) for n = 1..10000

Formula

a(n) = - A001221(n)*A008836(n).

a(n) = A001221(n)*(-1)^A073093(n).

a(n) = Sum_{d|n} mu(d)*A358769(n/d).

a(n) = Sum_{d|n} lambda(d)*A010051(n/d), where lambda = A008836.

a(n) = Sum_{d|n} omega(d)*A158522(n/d).

a(n) = Sum_{d|n} A010052(d)*A143519(n/d).

a(n) = Sum_{d|n} A062799(d)*A326415(n/d).

a(n) = Sum_{d|n} A007427(d)*A345345(n/d).

a(n*m) = lambda(n)*a(m) + lambda(m)*a(n), for all n, m such that gcd(n,m) = 1.

Dirichlet generating function: zeta(2s)/zeta(s) * primezeta(s).

Example

a(33) = omega(33)*(-1)^(bigomega(33)+1) = 2*(-1) = -2.

Maple

with(numtheory): seq(nops(factorset(n))*(-1)^(bigomega(n)+1), n=1..120);

Mathematica

Table[PrimeNu[n]*(-1)^(PrimeOmega[n] + 1), {n, 1, 120}]

Crossrefs

Cf. A001221, A008836, A358769, A073093, A008683, A158522.

Cf. A062799, A326415, A007427, A345345, A010052, A143519.

Cf. A158210, A246551, A056798, A187039, A279458.

Sequence in context: A237353 A293460 A231813 * A158210 A322307 A087802

Adjacent sequences: A392282 A392283 A392284 * A392286 A392287 A392288

Keyword

sign

Author

Ridouane Oudra, Feb 05 2026

Status

approved