The unitary divisor function 
, for a positive
integer 
,
is the analog of the divisor function 
for unitary divisors
and denotes the sum-of-
th-powers-of-the-unitary divisors
function. As in the case of the usual divisor function,
is commonly written 
.
The number of unitary divisors 
equals the number of squarefree divisors of 
, and is also 
, where 
is the number of distinct primes
dividing 
.
If 
is squarefree, then 
.
can be computed using the formula
 |
where the 
are distinct primes and the 
are positive integers.
It can be computed in the Wolfram Language
as
UnitaryDivisorSigma[k_, 1] = 1;
UnitaryDivisorSigma[k_, n_Integer] /; n > 1 := Times @@
(1 + (Power @@@ FactorInteger[n])^k)
The following table gives 
for 
, 2, ... and small 
.
 | OEIS |  for  , 2, ... |
| 0 | A034444 | 1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, ... |
| 1 | A034448 | 1, 3, 4, 5, 6, 12, 8, 9, 10, 18, 12, 20, 14, 24, 24, ... |
| 2 | A034676 | 1, 5, 10, 17, 26, 50, 50, 65, 82, 130, 122, 170, 170, 250, 260, ... |
| 3 | A034677 | 1, 9, 28, 65, 126, 252, 344, 513, 730, 1134, 1332, 1820, ... |
| 4 | A034678 | 1, 17, 82, 257, 626, 1394, 2402, 4097, 6562, 10642, ... |
