TOPICS
Search

Infinitary Divisor


p^x is an infinitary divisor of p^y (with y>0) if p^x|_(y-1)p^y, where d|_kn 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. Compute the prime factorization for each divisor d of n,

 d=product_(i=1)^kp_i^(alpha_i).

Now make a table of the binary representations (alpha_i)_2 of alpha_i for each prime factor p_i. The infinitary divisors are then those factors d that have zeros in the binary representation of all alpha_is where n itself does. This is illustrated in the following table for the number n=12, which has divisors 1, 2, 3, 4, 6, and 12 and prime factors 2 and 3.

dp_1alpha_1(alpha_1)_2p_2alpha_2(alpha_2)_2
12000030000
22100130000
32000031001
42201030000
62100131001
122201031001

As can be seen from the table, the divisors 1, 3, 4, and 12 have zeros in the binary expansions of alpha_1 (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 alpha_2 (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).

nd|_inftyn
11
21, 2
31, 3
41, 4
51, 5
61, 2, 3, 6
71, 7
81, 2, 4, 8
91, 9
101, 2, 5, 10
111, 11
121, 3, 4, 12
131, 13
141, 2, 7, 14
151, 3, 5, 15

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


See also

Divisor, Infinitary Perfect Number, k-ary Divisor, Unitary Divisor

Explore with Wolfram|Alpha

References

Abbott, P. "In and Out: k-ary Divisors." Mathematica J. 9, 702-706, 2005.Cohen, G. L. "On an Integer's Infinitary Divisors." Math. Comput. 54, 395-411, 1990.Cohen, G. and Hagis, P. "Arithmetic Functions Associated with the Infinitary Divisors of an Integer." Internat. J. Math. Math. Sci. 16, 373-383, 1993.Guy, R. K. Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 54, 1994.Sloane, N. J. A. Sequences A037445 and A077609 in "The On-Line Encyclopedia of Integer Sequences."

Referenced on Wolfram|Alpha

Infinitary Divisor

Cite this as:

Weisstein, Eric W. "Infinitary Divisor." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/InfinitaryDivisor.html

Subject classifications