A174863 Little omega analog to Liouville's function L(n).

1, 0, -1, -2, -3, -2, -3, -4, -5, -4, -5, -4, -5, -4, -3, -4, -5, -4, -5, -4, -3, -2, -3, -2, -3, -2, -3, -2, -3, -4, -5, -6, -5, -4, -3, -2, -3, -2, -1, 0, -1, -2, -3, -2, -1, 0, -1, 0, -1, 0, 1, 2, 1, 2, 3, 4, 5, 6, 5, 4, 3, 4, 5, 4, 5, 4, 3, 4, 5, 4, 3, 4, 3, 4, 5, 6, 7

Offset: 1

Author

Alonso del Arte, Dec 01 2010

Keywords

• sign

Comments

Instead of using the Omega function, number of prime factors counted with multiplicity, this is using the omega function, number of distinct prime factors.

Except for the two zeros and the intervening foray into negative territory shown here, the first thousand terms are all positive. The next zero occurs at term 7960. After the zero at term 12100, the function stays negative until term 22395666.

This sequence and the Liouville sequence have some terms up to a(43) exactly the same. I don't know at what higher point (if any) that is the case again. [del Arte]

It appears certain that this sequence and the Liouville sequence are equal infinitely often. Because they have the same parity and always change by one, they cannot cross without meeting. Both change signs infinitely often, and at apparently unrelated points. - Franklin T. Adams-Watters, Aug 05 2011

Examples

			a(4) = -2 because: a(1) = 1, as 1 has an even number of prime factors; then 2 and 3 being prime, bring the running sum down to -1; and then 4, which has one distinct prime factor, brings the sum down to -2. (This is the first term that differs from the Mertens function and Liouville's function.)
		

Links

• Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
• Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.
• Wolfgang Schwarz, A remark on multiplicative functions, Bulletin of the London Mathematical Society, Vol. 4, No. 2 (1972), pp. 136-140; alternative link.
• Jan van de Lune and Robert E. Dressler, Some theorems concerning the number theoretic function omega(n), Journal für die reine und angewandte Mathematik, Vol. 277 (1975), pp. 117-119; alternative link.

Crossrefs

Partial sums of A076479. - Reinhard Zumkeller, Jun 01 2013

Cf. A002819 (Liouville's function), A002321 (Mertens's function), A275547 (where a(n) is zero).

Cf. A346617.

Programs

Haskell

a174863 n = a174863_list !! (n-1)
a174863_list = scanl1 (+) a076479_list
-- Reinhard Zumkeller, Jun 01 2013

Mathematica

s=0; Table[s=s+(-1)^PrimeNu[n]; s, {n, 100}] (* PrimeNu is new in Mathematica 7.0 *)

PARI

a(n)=sum(k=1,n,(-1)^omega(k)) \\ Charles R Greathouse IV, Mar 27 2012

PARI

a(n)=my(v=vectorsmall(n, i, 1)); forprime(p=2, n, forstep(i=p, n, p, v[i]*=-1)); sum(i=1, #v, v[i]) \\ Charles R Greathouse IV, Aug 21 2016

Python

from sympy import primefactors
def omega(n): return 0 if n==1 else len(primefactors(n))
def a(n): return sum([(-1)**omega(i) for i in range(1, n + 1)]) # Indranil Ghosh, May 20 2017

Formula

a(n) = Sum_{i = 1..n} (-1)^omega(i).

A275547(a(n)) = 0. - Alois P. Heinz, Aug 03 2016

From Ridouane Oudra, Dec 31 2020: (Start)

a(n) = Sum_{i=1..n} Sum_{j=1..n} mu(i*j)*floor(n/(i*j));

a(n) = Sum_{i=1..n} mu(i)*tau(i)*floor(n/i);

a(n) = Sum_{i=1..n} 2^Omega(i)*mu(i)*floor(n/i), where Omega = A001222. (End)

From Amiram Eldar, Mar 05 2021: (Start)

a(n) ~ O(n * exp(-c*sqrt(log(n)))) (Schwarz, 1972).

a(n) ~ o(n) (van de Lune and Dressler, 1975). (End)

a(n) = 1 + Sum_{k>=1} (-1)^k * A346617(n,k). - Alois P. Heinz, Aug 19 2021