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.

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

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