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A174863
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Little omega analog to Liouville's function L(n).
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10
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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
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OFFSET
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1,4
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COMMENTS
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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
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LINKS
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Michael J. Mossinghoff and Timothy S. Trudgian, A tale of two omegas, arXiv:1906.02847 [math.NT], 2019.
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FORMULA
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a(n) = Sum_{i = 1..n} (-1)^omega(i).
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)
a(n) ~ O(n * exp(-c*sqrt(log(n)))) (Schwarz, 1972).
a(n) ~ o(n) (van de Lune and Dressler, 1975). (End)
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EXAMPLE
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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.)
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MATHEMATICA
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s=0; Table[s=s+(-1)^PrimeNu[n]; s, {n, 100}] (* PrimeNu is new in Mathematica 7.0 *)
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PROG
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(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
(Haskell)
a174863 n = a174863_list !! (n-1)
a174863_list = scanl1 (+) a076479_list
(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
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CROSSREFS
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KEYWORD
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sign
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AUTHOR
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STATUS
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approved
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