Unitary divisor

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In mathematics, a natural number a is a unitary divisor (or Hall divisor) of a number b if a is a divisor of b and if a and are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number.

Equivalently, a given divisor a of b is a unitary divisor if and only if every prime factor of a has the same multiplicity in a as it has in b.

The sum of unitary divisors function is denoted by the lowercase Greek letter sigma thus: σ*(n). The sum of the k-th powers of the unitary divisors is denoted by σ*k(n):

If the proper unitary divisors of a given number add up to that number, then that number is called a unitary perfect number.


The number of unitary divisors of a number n is 2k, where k is the number of distinct prime factors of n.

This is because each integer N > 1 is the product of positive powers prp of distinct prime numbers p. Thus every unitary divisor of N is the product, over a given subset S of the prime divisors {p} of N, of the prime powers prp for p ∈ S. If there are k prime divisors, then there are exactly 2k subsets S, and the statement follows.

The sum of the unitary divisors of n is odd if n is a power of 2 (including 1), and even otherwise.

Both the count and the sum of the unitary divisors of n are multiplicative functions of n that are not completely multiplicative. The Dirichlet generating function is

Every divisor of n is unitary if and only if n is square-free.

Odd unitary divisors[edit]

The sum of the k-th powers of the odd unitary divisors is

It is also multiplicative, with Dirichlet generating function

Bi-unitary divisors[edit]

A divisor d of n is a bi-unitary divisor if the greatest common unitary divisor of d and n/d is 1. The number of bi-unitary divisors of n is a multiplicative function of n with average order where[1]

A bi-unitary perfect number is one equal to the sum of its bi-unitary aliquot divisors. The only such numbers are 6, 60 and 90.[2]

OEIS sequences[edit]


  1. ^ Ivić (1985) p.395
  2. ^ Sandor et al (2006) p.115
  • Richard K. Guy (2004). Unsolved Problems in Number Theory. Springer-Verlag. p. 84. ISBN 0-387-20860-7. Section B3.
  • Paulo Ribenboim (2000). My Numbers, My Friends: Popular Lectures on Number Theory. Springer-Verlag. p. 352. ISBN 0-387-98911-0.
  • Cohen, Eckford (1959). "A class of residue systems (mod r) and related arithmetical functions. I. A generalization of Möbius inversion". Pacific J. Math. 9 (1): 13–23. doi:10.2140/pjm.1959.9.13. MR 0109806.
  • Cohen, Eckford (1960). "Arithmetical functions associated with the unitary divisors of an integer". Mathematische Zeitschrift. 74: 66–80. doi:10.1007/BF01180473. MR 0112861.
  • Cohen, Eckford (1960). "The number of unitary divisors of an integer". American Mathematical Monthly. 67 (9): 879–880. doi:10.2307/2309455. JSTOR 2309455. MR 0122790.
  • Cohen, Graeme L. (1990). "On an integers' infinitary divisors". Math. Comp. 54 (189): 395–411. Bibcode:1990MaCom..54..395C. doi:10.1090/S0025-5718-1990-0993927-5. MR 0993927.
  • Cohen, Graeme L. (1993). "Arithmetic functions associated with infinitary divisors of an integer". Int. J. Math. Math. Sci. 16 (2): 373–383. doi:10.1155/S0161171293000456.
  • Finch, Steven (2004). "Unitarism and Infinitarism" (PDF).
  • Ivić, Aleksandar (1985). The Riemann zeta-function. The theory of the Riemann zeta-function with applications. A Wiley-Interscience Publication. New York etc.: John Wiley & Sons. p. 395. ISBN 0-471-80634-X. Zbl 0556.10026.
  • Mathar, R. J. (2011). "Survey of Dirichlet series of multiplicative arithmetic functions". arXiv:1106.4038 [math.NT]. Section 4.2
  • Sándor, József; Mitrinović, Dragoslav S.; Crstici, Borislav, eds. (2006). Handbook of number theory I. Dordrecht: Springer-Verlag. ISBN 1-4020-4215-9. Zbl 1151.11300.
  • Toth, L. (2009). "On the bi-unitary analogues of Euler's arithmetical function and the gcd-sum function". J. Int. Seq. 12.

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