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 A327666 a(n) = Sum_{k = 1..n} (-1)^(Omega(k) - omega(k)), where Omega(k) counts prime factors of k with multiplicity and omega(k) counts distinct prime factors. 2
 1, 2, 3, 2, 3, 4, 5, 6, 5, 6, 7, 6, 7, 8, 9, 8, 9, 8, 9, 8, 9, 10, 11, 12, 11, 12, 13, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 26, 25, 26, 27, 26, 25, 24, 25, 24, 25, 26, 27, 28, 29, 30, 31, 30, 31, 32, 31, 30, 31, 32, 33, 32, 33, 34 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Partial sums of A162511. LINKS Amiram Eldar, Table of n, a(n) for n = 1..10000 FORMULA a(1) = 1, a(n) = a(n - 1) + (-1)^(Omega(n) - omega(n)) for n > 1. a(n) ~ c * n, where c = A307868. - Amiram Eldar, Sep 18 2022 EXAMPLE Omega(1) = omega(1) = 0. The difference is 0, so (-1)^0 = 1, so a(1) = 1. Omega(2) = omega(2) = 1. The difference is 0, so (-1)^0 = 1, which is added to a(1) to give a(2) = 2. Omega(3) = omega(3) = 1. The difference is 0, so (-1)^0 = 1, which is added to a(2) to give a(3) = 3. Omega(4) = 2 but omega(4) = 1. The difference is 1, so (-1)^1 = -1, which is added to a(3) to give a(4) = 2. MATHEMATICA Table[Sum[(-1)^(PrimeOmega[k] - PrimeNu[k]), {k, n}], {n, 70}] f[p_, e_] := (-1)^(e - 1); Accumulate @ Table[Times @@ f @@@ FactorInteger[n], {n, 1, 100}] (* Amiram Eldar, Sep 18 2022 *) PROG (PARI) seq(n)={my(v=vector(n)); v[1]=1; for(k=2, n, v[k] = v[k-1] + (-1)^(bigomega(k)-omega(k))); v} \\ Andrew Howroyd, Sep 23 2019 (Python) from functools import reduce from sympy import factorint def A327666(n): return sum(-1 if reduce(lambda a, b:~(a^b), factorint(i).values(), 0)&1 else 1 for i in range(1, n+1)) # Chai Wah Wu, Jan 01 2023 CROSSREFS Cf. A001221, A001222, A002321, A002819, A008836, A046660, A069812, A076479, A162511, A174863, A307868. Sequence in context: A289172 A215653 A358503 * A286717 A162751 A026342 Adjacent sequences: A327663 A327664 A327665 * A327667 A327668 A327669 KEYWORD nonn AUTHOR Ilya Gutkovskiy, Sep 21 2019 STATUS approved

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Last modified August 9 18:14 EDT 2024. Contains 375044 sequences. (Running on oeis4.)