10.23 Tau

Tau( n )

Tau returns the number of the positive divisors (see DivisorsInt) of the integer n.

Tau is a multiplicative arithmetic function, i.e., if n and m are relatively prime we have tau(n m) = tau(n) tau(m). Together with the formula tau(p^e) = e+1 this allows us to compute tau(n).

Tau usually spends most of its time factoring n (see FactorsInt).

    gap> Tau( 0 );
    Error, Tau: <n> must not be 0
    gap> Tau( 1 );
    1
    gap> Tau( 1013 );
    2        # thus 1013 is a prime
    gap> Tau( 8128 );
    14
    gap> Tau( 36 );
    9        # $\tau(n)$ is odd if and only if $n$ is a perfect square 

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