Infinitary Divisor

An infinitary divisor of a prime power , where is prime and is a positive integer, is a divisor satisfying . Here, denotes a k-ary divisor (Guy 1994, p. 54). Infinitary divisors therefore generalize the concept of the k-ary divisor.

Infinitary divisors can also be defined as follows. For a positive integer , compute the prime factorization for each divisor of ,

Now make a table of the binary representations of for each prime factor . The infinitary divisors are then those factors that have zeros in the binary representation of all s where itself does. This is illustrated in the following table for the number , which has divisors 1, 2, 3, 4, 6, and 12 and prime factors 2 and 3.

1	2	0	000	3	0	000
2	2	1	001	3	0	000
3	2	0	000	3	1	001
4	2	2	010	3	0	000
6	2	1	001	3	1	001
12	2	2	010	3	1	001

As can be seen from the table, the divisors 1, 3, 4, and 12 have zeros in the binary expansions of (the exponents of 2) in the positions that 12 itself does. Similarly, all divisors have zeros in the leftmost two positions in the binary expansions of (the exponents of 3), as does 12 itself. The intersection of the divisors matching zero in the binary representations in each of the exponents is therefore 1, 3, 4, 12, and these are the infinitary divisors of 12.

The following table lists the infinitary divisors for small integers (OEIS A077609).

1	1
2	1, 2
3	1, 3
4	1, 4
5	1, 5
6	1, 2, 3, 6
7	1, 7
8	1, 2, 4, 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

The numbers of infinitary divisors of for , 2, ... are 1, 2, 2, 2, 2, 4, 2, 4, 2, 4, ... (OEIS A037445).