Unitary divisor

In mathematics, a natural number a is a unitary divisor of a number b if a is a divisor of b and if a and $\frac{b}{a}$ are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and $\frac{60}{5}=12$ have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and $\frac{60}{6}=10$ 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):

$\sigma_k^*(n) = \sum_{d\mid n \atop \gcd(d,n/d)=1} \!\! d^k.$

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

Properties

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

$\frac{\zeta(s)\zeta(s-k)}{\zeta(2s-k)} = \sum_{n\ge 1}\frac{\sigma_k^*(n)}{n^s}.$

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

Odd unitary divisors

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

$\sigma_k^{(o)*}(n) = \sum_{{d\mid n \atop d\equiv 1 \pmod 2} \atop \gcd(d,n/d)=1} \!\! d^k.$

It is also multiplicative, with Dirichlet generating function

$\frac{\zeta(s)\zeta(s-k)(1-2^{k-s})}{\zeta(2s-k)(1-2^{k-2s})} = \sum_{n\ge 1}\frac{\sigma_k^{(o)*}(n)}{n^s}.$

Bi-unitary divisors

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 $A \log x$ where^[1]

$A = \prod_p\left({1 - \frac{p-1}{p^2(p+1)} }\right) \ .$

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]

References

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). pp. 13—23. MR 0109806.
• Cohen, Eckford (1960). "Arithmetical functions associated with the unitary divisors of an integer". Mathematische Zeitschrift 74. pp. 66—80. doi:10.1007/BF01180473. MR 0112861.
• Cohen, Eckford (1960). "The number of unitary divisors of an integer". American mathematical monthly 67 (9). pp. 879—880. MR 0122790.
• Cohen, Graeme L. (1990). "On an integers' infinitary divisors". Math. Comp. 54 (189). pp. 395—411. doi:10.1090/S0025-5718-1990-0993927-5. MR 0993927.
• Cohen, Graeme L. (1993). "Arithmetic functions associated with infinitary divisors of an integer". Intl. J. Math. Math. Sci. 16 (2). pp. 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.
• 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.

External links

• Weisstein, Eric W., "Unitary Divisor", MathWorld.

OEIS sequences

•  A034444 is σ₀(n) 
•   A034448 is σ₁(n) 
•   A034676 to  A034682 are σ₂(n) to σ₈(n) 
•   A068068 is σ^(o)*₀(n) 
•   A192066 is σ^(o)*₁(n)