Unitary Divisor Function

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, ...

REFERENCES:

Sloane, N. J. A. Sequences A034444, A034448, A034676, A034677, and A034678 in "The On-Line Encyclopedia of Integer Sequences."