BUILT-IN MATHEMATICA SYMBOL

EulerPhi

EulerPhi[n]
gives the Euler totient function

.

MORE INFORMATION

Integer mathematical function, suitable for both symbolic and numerical manipulation.

gives the number of positive integers less than or equal to n which are relatively prime to n.

is taken to be equal to .

EulerPhi automatically threads over lists.

EXAMPLES

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Basic Examples (2)

Table of values:

Plot the sequence:

Table of values:

In[1]:=

Out[1]=

Plot the sequence:

In[1]:=

Out[1]=

Scope (3)

Evaluate for large arguments:

EulerPhi threads element-wise over lists:

TraditionalForm formatting:

Generalizations & Extensions (1)

EulerPhi works for negative integers:

Applications (8)

A surprising result for powers of 10:

Number of order-n Farey fractions with denominator n:

Plot the cumulative sum of EulerPhi:

Compare with an asymptotic approximation:

First several

-s where the difference

is negative:

The probability that two randomly chosen positive integers less than x are relatively prime:

Compare with the asymptotic limit:

Number of cyclic necklaces of length n that can be formed with b types of beads:

Power series of the generating function for GCD:

Count the number of primes using EulerPhi:

Model Fleck's totient function:

reproduces the Euler totient function:

Generalizations and closed forms:

Properties & Relations (4)

Use FullSimplify with assumptions to simplify expressions containing EulerPhi:

EulerPhi is a multiplicative function:

Calculate EulerPhi through prime factorization:

Calculate a generating function of EulerPhi:

Possible Issues (1)

Value at 0:

Neat Examples (3)

Form an absolutely abnormal number as the limit of the following sequence:

Digits of the sixth approximation in various bases:

Iterate the map

and display result modulo

:

The only 8 solutions of

: