A353338 -id:A353338 - cp's OEIS frontend

A065043 Characteristic function of the numbers with an even number of prime factors (counted with multiplicity): a(n) = 1 if n = A028260(k) for some k then 1 else 0.

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 0

Offset: 1

Reinhard Zumkeller, Nov 05 2001

nonn

Antti Karttunen, Table of n, a(n) for n = 1..65537 (first 1000 terms from Harry J. Smith)
Index entries for characteristic functions
Index entries for sequences computed from exponents in factorization of n

Characteristic function of A028260 (positions of 1's). Cf. also A026424 (positions of 0's) and A320655.
One less than A007421.
Cf. A003961, A008836, A010052, A038548 (inverse Möbius transform), A046523, A055037 (partial sums), A343784, A347102, A353337, A353338, A353555, A353557, A353629, A353669, A358750, A358752, A353374, A358775, A356163, A356170.
Cf. also A066829, A353374.

A065043 := proc(n)
    if type(numtheory[bigomega](n),'even') then
        1;
    else
        0;
    end if;
end proc: # R. J. Mathar, Jun 26 2013

Table[(LiouvilleLambda[n]+1)/2,{n,1,20}] (* Enrique Pérez Herrero, Jul 07 2012 *)

{ for (n=1, 1000, a=1 - bigomega(n)%2; write("b065043.txt", n, " ", a) ) } \\ Harry J. Smith, Oct 04 2009

A065043(n) = (1 - (bigomega(n)%2)); \\ Antti Karttunen, Apr 19 2022

from operator import ixor
from functools import reduce
from sympy import factorint
def A065043(n): return (reduce(ixor, factorint(n).values(),0)&1)^1 # Chai Wah Wu, Jan 01 2023

a(n) = 1 - A001222(n) mod 2.
a(n) = A007421(n) - 1.
a(n) = 1 - A066829(n).
a(A028260(k)) = 1 and a(A026424(k)) = 0 for all k.
Dirichlet g.f.: (zeta(s)^2 + zeta(2*s))/(2*zeta(s)). - Enrique Pérez Herrero, Jul 06 2012
a(n) = (A008836(n) + 1)/2. - Enrique Pérez Herrero, Jul 07 2012
a(n) = A001222(2n) mod 2. - Wesley Ivan Hurt, Jun 22 2013
G.f.: Sum_{n>=1} a(n)*x^n/(1 - x^n) = Sum_{n>=1} x^(n^2)/(1 - x^n). - Ilya Gutkovskiy, Apr 25 2017
From Antti Karttunen, Dec 01 2022: (Start)
For x, y >= 1, a(x*y) = 1 - abs(a(x)-a(y)).
a(n) = a(A046523(n)) = A356163(A003961(n)).
a(n) = A000035(A356163(n)+A347102(n)).
a(n) = A010052(n) + A353669(n).
a(n) = A353555(n) + A353557(n).
a(n) = A358750(n) + A358752(n).
a(n) = A353374(n) + A358775(n).
a(n) >= A356170(n).
(End)

Corrected by Charles R Greathouse IV, Sep 02 2009

A353337 Number of ways to write n as a product of the terms of A028260 larger than 1; a(1) = 1 by convention (an empty product).

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 0, 1, 1, 0, 0, 0, 1, 1, 2, 0, 0, 0, 0, 1, 1, 0, 2, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 3, 0, 1, 1, 2, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 2, 1, 2, 1, 1, 0, 3, 0, 1, 0, 3, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 3, 1, 1, 1, 2, 0, 3, 1, 0, 1, 1, 1, 4, 0, 0, 0, 3, 0, 0, 0, 2, 0

Offset: 1

Antti Karttunen, Apr 17 2022

nonn

Number of factorizations of n into factors k > 1 for which there is an even number of primes (when counted with multiplicity, A001222) in their prime factorization.

			Of the eleven divisors of 96 larger than one, the following: [4, 6, 16, 24, 96] are terms of A028260 because they have an even number of prime factors when counted with repetition. Using them, we can factor 96 in four possible ways, as 96 = 24*4 = 16*6 = 6*4*4, therefore a(96) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n

Cf. A001222, A028260, A038548, A046523, A065043, A353338 [= a(n^2)].

Cf. also A320655, A353377.

A065043(n) = (1 - (bigomega(n)%2));
A353337(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A065043(d), s += A353337(n/d, d))); (s));

a(n) = a(A046523(n)). [The sequence depends only on the prime signature of n].

For all n >= 1, a(n) >= A320655(n), and a(n) >= A353377(n).

A353378 Number of ways to write the square of n as a product of the terms of A345452 larger than 1; a(1) = 1 by convention (an empty product).

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 3, 5, 1, 4, 1, 4, 3, 2, 1, 7, 2, 2, 3, 4, 1, 7, 1, 7, 3, 2, 3, 9, 1, 2, 3, 7, 1, 7, 1, 4, 6, 2, 1, 12, 2, 4, 3, 4, 1, 7, 3, 7, 3, 2, 1, 16, 1, 2, 6, 11, 3, 7, 1, 4, 3, 7, 1, 16, 1, 2, 6, 4, 3, 7, 1, 12, 5, 2, 1, 16, 3, 2, 3, 7, 1, 17, 3, 4, 3, 2, 3, 19, 1, 4, 6, 9, 1, 7

Offset: 1

Antti Karttunen, Apr 17 2022

nonn

Number of factorizations of n^2 into factors k > 1 for which there is an even number of primes (when counted with multiplicity, A001222) in their prime factorization, and the 2-adic valuation of k (A007814) is also even.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000290, A001222, A007814, A345452, A353374, A353377.

Cf. also A353304, A353334, A353338, A353359.

A353374(n) = (!(bigomega(n)%2) && !(valuation(n, 2)%2));
A353377(n, m=n) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1)&&(d<=m)&&A353374(d), s += A353377(n/d, d))); (s));
A353378(n) = A353377(n^2);

a(n) = A353377(A000290(n)).

For all n >= 1, a(n) <= A353338(n).

cp's OEIS Frontend

A065043 Characteristic function of the numbers with an even number of prime factors (counted with multiplicity): a(n) = 1 if n = A028260(k) for some k then 1 else 0.

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A353337 Number of ways to write n as a product of the terms of A028260 larger than 1; a(1) = 1 by convention (an empty product).

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A353378 Number of ways to write the square of n as a product of the terms of A345452 larger than 1; a(1) = 1 by convention (an empty product).

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