Parity of (unique) number of prime factors

Question

Analysis on the parity of the number of prime factors of an integer $n$ was performed. Here I displayed the parity as an random walk till $n=100.000.000$:

• Parity of the total number of prime factors (including duplicates). Step: y=+1 (odd), y=-1 (even)
• Parity of the total number of unique prime factors. Step: x=+1 (odd), x=-1 (even)

Example: $3^5 \cdot 7 \cdot 13^2$

Sum of unique divisors: $3$ is odd so: $y=+1$.

Sum of exponents (total number divisors): $8$ is even so: $x=-1$.

I did some investigation if both these series are bounded or have a grow rate. Though are not able to understand correctly. I found a question on SE and information at Wiki.

According question on SE:

See also problem 168307 (Generalized PNT in limit as numbers get large) which shows that in a sense even numbers have more prime factors than odd ones, and cites a result of Ramanujan that the "normal order" of prime factors of a number n, with or without repetitions, is about loglogn. https://math.stackexchange.com/a/315791/650339

Wiki Parity problem:

Parity problem. If A is a set whose elements are all products of an odd number of primes (or are all products of an even number of primes), then (without injecting additional ingredients), sieve theory is unable to provide non-trivial lower bounds on the size of A. Also, any upper bounds must be off from the truth by a factor of 2 or more. https://en.wikipedia.org/wiki/Parity_problem_(sieve_theory)

Questions:

1. Are both quotes related to the same problem? I am just an amateur trying to get a better understanding.
2. Will the parity of all prime factors (including duplicates) always tend to be ODD? That is the y-axis in the graph and remains positive till now.

Random walk: Parity of (unique) number of prime factors.

Movie Random walk growth: https://youtu.be/zfFp8p94FG4

import numpy as np
import matplotlib.pyplot as plt1
from sympy.ntheory import factorint
from matplotlib.ticker import (MultipleLocator, FormatStrFormatter,
                               AutoMinorLocator)

%matplotlib widget

fig, ax1a = plt1.subplots(1, figsize=(16, 8))
ax1a.clear()

#Number of numbers to factorize
p=100000

#maximum plotpoints in graph, recommended <2.000.000
s=2000000

if (s>=p):
    s=p

modu=int(p/s)

#Set counter zero and create zero arrays plot data
c=0
xcs=np.zeros(s+1)
ycs=np.zeros(s+1)
xs=np.zeros(s+1)
ys=np.zeros(s+1)

#Images to save in walk (parital walk, red), images=0 no partial random walks
images=5

#Set cumulative biffer value to 0
xm=0
ym=0

#Spacing tickmarks x and y axis
tick=2000

for n in range(p+1):
    
    #Prime Factors
    a=factorint(n)
    
    #Unique Prime Factors
    pf=list(a)

    #All Exponents
    exp=list(a.values())
    
    #Count all Prime Factors (including duplicates) if even -1 odd +1
    if np.sum(exp)%2==0:
        yn=-1
    else:
        yn=1
    
    #Add results to array, including cumulative steps
    ym=ym+yn

    #Count all unique Prime Factors (including duplicates) if even -1 odd +1
    pf=np.size(pf)
    if pf%2==0:
        xn=-1
    else:
        xn=1

    #Add results to array, including cumulative steps
    xm=xm+xn 

    #Add results to shortened table with maximum of s elements
    if (n%modu==0):
        xcs[c]=xm
        ycs[c]=ym

        xs[c]=xn
        ys[c]=yn

        c=c+1
        
        
#Plot total random walk
ax1a.plot(xcs,ycs, marker='', color='black', linestyle='-', markersize=0,linewidth=0.04)
ax1a.set_xlabel('$x$',fontsize=10)
ax1a.set_ylabel('$y$',fontsize=10)
ax1a.grid(b=True, which='major', color='#666666', linestyle='-', zorder=0)
ax1a.ticklabel_format(axis='x', style='sci', scilimits=None, useOffset=None, useLocale=None, useMathText=None)
ax1a.ticklabel_format(axis='y', style='sci', scilimits=None, useOffset=None, useLocale=None, useMathText=None)
ax1a.xaxis.set_major_locator(MultipleLocator(tick))
ax1a.yaxis.set_major_locator(MultipleLocator(tick))

plt1.title('Count all factors:  y=+1 (odd), y=-1 (even), Count unique factors: x=+1 (odd), x=-1 (even)', fontsize=10)
plt1.suptitle('Parity random walk of prime factorization', fontsize=25,y=0.96)
plt1.gca().set_aspect('equal', adjustable='box')
plt1.savefig('Random Walk00000.png', dpi=300, bbox_inches='tight')

if images>0:
    for q in range(images):

        count=int((q+1)*s/images)

        #print(count)    
        a=factorint(int(count*p/s))


        #Unique Prime Factors
        pf=np.char.array(np.array(list(a),dtype=str))

        #Exponents, create Latex string with primefactors and exponents
        exp=np.char.array(np.array(list(a.values()),dtype=str))
        fac='$' + (', '.join(pf+'^{' + exp + '}')) + '$'

        #Create graph label for last point partial random walk
        l='n= ' + str(int(count*p/s)) + '\nfactors: ' + str(fac) + '\nparity: x=' +  str(xs[(count)]) + ', y=' +  str(ys[(count)]) + '\ncumulative: x=' +  str(xcs[(count)]) + ', y=' +  str(ycs[(count)]) 

        #Plot black total walk and partial red 
        ax1a.clear()
        ax1a.plot(xcs,ycs, marker='', color='black', linestyle='-', markersize=0,linewidth=0.04)
        #ax1a.plot(xcs[0:(count)],ycs[0:(count)], marker='', color='red', linestyle='-', markersize=0,linewidth=0.05)
        ax1a.plot(xcs[(count)],ycs[(count)], marker='o', color='red', linestyle='-', markersize=3,linewidth=0,label=l)

        ax1a.grid(b=True, which='major', color='#666666', linestyle='-', zorder=0)
        ax1a.xaxis.set_major_locator(MultipleLocator(tick))
        ax1a.yaxis.set_major_locator(MultipleLocator(tick))
        ax1a.ticklabel_format(axis='x', style='sci', scilimits=None, useOffset=None, useLocale=None, useMathText=None)
        ax1a.ticklabel_format(axis='y', style='sci', scilimits=None, useOffset=None, useLocale=None, useMathText=None)
        ax1a.legend(loc='upper left',fontsize=12,markerscale=1,frameon=False)
        ax1a.set_xlabel('$x$',fontsize=10)
        ax1a.set_ylabel('$y$',fontsize=10)

        plt1.title('Count all factors:  y=+1 (odd), y=-1 (even), Count unique factors: x=+1 (odd), x=-1 (even)', fontsize=10)
        plt1.suptitle('Parity random walk of prime factorization', fontsize=25,y=0.96)
        plt1.gca().set_aspect('equal', adjustable='box')
        #plt1.savefig('Random Walk' + str("%05d" % (q+1)) +'.png', dpi=300, bbox_inches='tight')