Divisor Count of 111,111

Example of Use of Divisor Counting Function

 * $\map \tau {111 \, 111} = 32$

where $\tau$ denotes the divisor counting (tau) function.

Proof
From Divisor Counting Function from Prime Decomposition:
 * $\ds \map \tau n = \prod_{j \mathop = 1}^r \paren {k_j + 1}$

where:
 * $r$ denotes the number of distinct prime factors in the prime decomposition of $n$
 * $k_j$ denotes the multiplicity of the $j$th prime in the prime decomposition of $n$.

We have that:
 * $111 \, 111 = 3 \times 7 \times 11 \times 13 \times 37$

Thus:

The divisors of $105$ can be enumerated as:
 * $1, 3, 7, 11, 13, 21, 33, 37, 39, 77, 91, 111, 143, 231,$
 * $259, 273, 407, 429, 481, 777, 1001, 1221, 1443, 2849,$
 * $3003, 3367, 5291, 8547, 10101, 15873, 37037, 111111$