Wolfram Function Repository

Instant-use add-on functions for the Wolfram Language

Function Repository Resource:

 Primorial

Source Notebook

Evaluate the primorial

Contributed by: Sander Huisman

ResourceFunction["Primorial"][n]

calculates the product of the primes up to n.

Details

Integer mathematical function, suitable for both symbolic and numerical manipulation.
For n≤1, the primorial is equal to 1.
For n≥2, the primorial is defined as , where pk is the kth prime and π(n) is PrimePi[n].
ResourceFunction["Primorial"] has the attribute Listable.

Examples

Basic Examples (2) 

Compute the primorial :

In[1]:=
ResourceFunction["Primorial"][23]
Out[1]=

Compute a list of the first 15 primorials:

In[2]:=
ResourceFunction["Primorial"][Range[15]]
Out[2]=

Scope (1) 

For non-positive numbers, the primorial is defined to be equal to 1:

In[3]:=
ResourceFunction["Primorial"][-4]
Out[3]=

Properties and Relations (2) 

Compare with the definition:

In[4]:=
With[{k = 7}, Product[Prime[j], {j, 1, PrimePi[k]}] == ResourceFunction["Primorial"][k]]
Out[4]=

The resource function ChebyshevTheta is the logarithm of the primorial:

In[5]:=
With[{k = 7}, Exp[ResourceFunction["ChebyshevTheta"][k]] == ResourceFunction["Primorial"][k]]
Out[5]=

Neat Examples (3) 

Evaluate the infinite primorial:

In[6]:=
Product[Prime[k], {k, 1, \[Infinity]}, Regularization -> "Dirichlet"]
Out[6]=
In[7]:=
ResourceFunction["Primorial"][\[Infinity]]
Out[7]=

Compare the growth rate of the primorial to that of the factorial:

In[8]:=
DiscretePlot[{ResourceFunction["Primorial"][n], Factorial[n]}, {n, 1, 50}, ScalingFunctions -> "Log", PlotLegends -> {"n#", "n!"}]
Out[8]=

Plot the differences between the factorial and the primorial up to n:

In[9]:=
ListLinePlot[ Differences[ Table[Log[n!/ResourceFunction["Primorial"][n]], {n, 1, 100, 1}]], Filling -> Axis]
Out[9]=

Publisher

SHuisman

Version History

  • 1.0.0 – 17 August 2021

Related Resources

License Information