Unitary divisor

Abstract: We classify the gapped phases of ZN parafermions in one dimension and construct a representative of each phase. Even in the absence of additional symmetries besides parafermionic parity, parafermions may be realized in a variety of phases, one for each divisor n of N. The phases can be characterized by spontaneous symmetry breaking, topology, or a mixture of the two. Purely topological phases arise if n is a unitary divisor, i.e. if n and N/n are co-prime. Our analysis is based on the explicit realization of all symmetry broken gapped phases in the dual ZN-invariant quantum spin chains.

Abstract: The notions of unitary divisor and biunitary divisor are extended in a natural fashion to give k-ary divisors, for any natural number k. We show that we may sensibly allow k to increase indefinitely, and this leads to infinitary divisors. The infinitary divisors of an integer are described in full, and applications to the obvious analogues of the classical perfect and amicable numbers and aliquot sequences are given.

Abstract: Let d be a divisor of a positive integer n. Then d is a unitary divisor if d and nld are relatively prime, and d is a bi-unitary divisor if the greatest common unitary divisor of d and nld is 1. An integer is bi-unitaty perfect if it equals the sum of its proper biunitary divisors. The purpose of this paper is to show that there are only three bi-unitary perfect numbers, namely 6, 60 and 90. A divisor d of an integer n is a unitary divisor if d and nld are relatively prime. A divisor d of an integer n is a bi-unitary divisor if the greatest common unitary divisor of d and nld is 1. Let a(n) be the sum of the divisors of n, let ?*(n) be the sum of the unitary divisors of n, and let ?**(n) be the sum of the bi-unitary divisors of n. We say that N is unitary perfect if a * (N) = 2N. Subbarao and Warren [2] showed that 6, 60, 90 and 87360 are the first four unitary perfect numbers; Wall reported [3] that 146,361,946,186,458,562,560,000 = 2183.547.11.13.19.37.79.109.157.313 is also unitary perfect and later showed [4] it to be the next such number after 87360. Subbarao [1] has conjectured that there are only finitely many unitary perfect numbers. We say that N is bi-unitary perfect if a* * (N) = 2N. The purpose of this paper is to show that the first three unitary perfect numbers, i.e., 6, 60 and 90, are the only bi-unitary perfect numbers. One easily verifies that ov* is multiplicative and that if p is prime and e> 1, then rY**(pe) = ry(pe) = (pe+l l)/(p 1) if e is odd, and or**(pe) = (pe+l l)/(p 1) -pe/2 if e is even. Hence a**(n)?a(n) with equality if and only if every prime which divides n does so an odd number of times. It also follows immediately that a* * (n) is odd if and only if n is 1 or a power of 2; consequently, each odd prime power unitary divisor of n contributes at least one factor 2 to a**(n). Received by the editors June 10, 1971. AMS 1970 subject classifications. Primary lOA20; Secondary lOA99.

Abstract: Two positive integers are said to be unitary amicable if the sum of the unitary divisors of each is equal to their sum. In this paper a table of such numbers is given, and some theorems concerning them are proved. 1. Introduction. In this paper, lower-case letters will be used to denote positive integers, with p and q always representing primes. If cd = n and (c, d) = 1, then d is said to be a unitary divisor of n. We shall use the symbol v*(n) to denote the sum of the unitary divisors of n. It is immediate that v*(1) = 1, while