Collatz Sequences for pn, n!, and ♯(n)
Collatz Sequences for , , and
p
n
n!
♯(n)
Paul Abbott
School of Physics, Mathematics and Computing
The University of Western Australia
School of Physics, Mathematics and Computing
The University of Western Australia
Collatz sequence
Collatz sequence
Primes
Primes
Factorial
Factorial
Primorial
Primorial
The primorial is defined as the product of first prime numbers:
♯(n)
n
In[]:=
♯(n_):=
n
∏
k=1
p
k
Compute for :
♯(n)
1≤n≤10
In[]:=
♯/@Range[10]
Out[]=
{2,6,30,210,2310,30030,510510,9699690,223092870,6469693230}
By construction, each primorial is divisible by only one power .
2
Compute the sequence length for versus for :
♯(n)
log(♯(n))
1≤n≤1000
In[]:=
data=(n{Log[♯[n]],Length@cs[♯[n]]})/@Range[1000];
Display the data:
In[]:=
plp=ListPlot[data,ImageSizeLarge]
Out[]=
A linear fit works well:
In[]:=
fit=Fit[data,x,x]
Out[]=
6.95456x
In[]:=
Show[plp,Plot[fit,{x,0,8000},PlotStyleRed]]
Out[]=