Unitary Divisor

A divisor  of  for which

(1)

where  is the greatest common divisor. For example, the divisors of 12 are , so the unitary divisors are . A list of unitary divisors of a number  an be computed in the Wolfram Language using:

  UnitaryDivisors[n_Integer] := Sort[Flatten[Outer[
    Times, Sequence @@ ({1, #}& /@
      Power @@@ FactorInteger[n])
   ]]]

The following table gives the unitary divisors for the first few integers (OEIS A077610).

1	1
2	1, 2
3	1, 3
4	1, 4
5	1, 5
6	1, 2, 3, 6
7	1, 7
8	1, 8
9	1, 9
10	1, 2, 5, 10
11	1, 11
12	1, 3, 4, 12
13	1, 13
14	1, 2, 7, 14
15	1, 3, 5, 15

Given the prime factorization

(2)

then

(3)

is a unitary divisor of  if each  is 0 or . For a prime power , the unitary divisors are 1 and  (Cohen 1990).

The symbol  is used to denote to the unitary divisor function.

The numbers of unitary divisors  of , 2, ... are 1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 4, 2, 4, 4, 2, 2, 4, 2, 4, ... (OEIS A034444). These numbers are also the numbers of squarefree divisors of . The number of unitary divisors of  is also given by , where  is the number of different primes dividing .

REFERENCES:

Cohen, G. L. "On an Integer's Infinitary Divisors." Math. Comput. 54, 395-411, 1990.

Guy, R. K. "Unitary Perfect Numbers." §B3 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 53-59, 1994.

Sloane, N. J. A. Sequences A034444 and A077610 in "The On-Line Encyclopedia of Integer Sequences."