The unitary divisor function 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 numbers of unitary divisors is the same as the numbers of squarefree divisors of , as well as , where is the number of different primes dividing .
If is squarefree, then .
can be computed using the formula
which can be computed in the Wolfram Language as
UnitaryDivisorSigma[k_, n_Integer] := 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, ... |