A353337 -id:A353337 - 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

A353377 Number of ways to write 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, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 2, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 2, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 2, 0, 0, 0, 3, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 2, 0, 0, 2, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1

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, and the 2-adic valuation of k (A007814) is also even.

			Of the 19 divisors of 240 larger than 1, the following: [4, 15, 16, 60, 240] are found in A345452. Using them, we can factor 240 in four possible ways, as 240 = 60*4 = 16*15 = 15*4*4, therefore a(240) = 4.
Of the 23 divisors of 540 larger than 1, the following: [4, 9, 15, 36, 60, 135, 540] are found in A345452. Using them, we can factor 540 in five possible ways, as 540 = 135*4 = 60*9 = 36*15 = 15*9*4, therefore a(540) = 5.

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

Cf. A001222, A007814, A345452, A353374, A353378 [= a(n^2)].

Cf. also A353333, A353337, A353353, A353373.

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));

For all n >= 1, a(n) <= A353337(n).

A353338 Number of ways to write the square of 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, 1, 1, 2, 1, 3, 1, 3, 2, 3, 1, 6, 1, 3, 3, 5, 1, 6, 1, 6, 3, 3, 1, 12, 2, 3, 3, 6, 1, 12, 1, 7, 3, 3, 3, 16, 1, 3, 3, 12, 1, 12, 1, 6, 6, 3, 1, 21, 2, 6, 3, 6, 1, 12, 3, 12, 3, 3, 1, 33, 1, 3, 6, 11, 3, 12, 1, 6, 3, 12, 1, 33, 1, 3, 6, 6, 3, 12, 1, 21, 5, 3, 1, 33, 3, 3, 3, 12, 1, 33, 3, 6, 3, 3, 3, 36, 1, 6, 6

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.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Index entries for sequences computed from exponents in factorization of n

Cf. A000290, A001222, A028260, A038548, A046523, A065043, A353337.

Cf. also A353378.

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));
A353338(n) = A353337(n^2);

a(n) = A353337(A000290(n)).
a(n) = a(A046523(n)). [The sequence depends only on the prime signature of n].
For all n >= 1, a(n) >= A353378(n).

A357374 Number of ordered factorizations of n into numbers > 1 with an even number of prime divisors (prime factors counted with multiplicity).

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, 3, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 4, 0, 1, 1, 3, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 1, 0, 0, 3, 1, 3, 1, 1, 0, 5, 0, 1, 0, 4, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 2, 1, 0, 5, 1, 1, 1, 3, 0, 5

Offset: 1

Ilya Gutkovskiy, Sep 25 2022

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A028260, A050334, A065043, A074206, A308063, A353337, A357375.

A065043[n_] := Boole[EvenQ[PrimeOmega[n]]]; a[1] = 1; a[n_] := a[n] = Sum[If[d < n, A065043[n/d] a[d], 0], {d, Divisors[n]}]; Table[a[n], {n, 1, 90}]

a(1) = 1; a(n) = Sum_{d|n, d < n} A065043(n/d) * a(d).

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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A353377 Number of ways to write n as a product of the terms of A345452 larger than 1; a(1) = 1 by convention (an empty product).

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A353338 Number of ways to write the square of n as a product of the terms of A028260 larger than 1; a(1) = 1 by convention (an empty product).

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A357374 Number of ordered factorizations of n into numbers > 1 with an even number of prime divisors (prime factors counted with multiplicity).

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Mathematica

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