- Title
- Square-free rank of integers
- Creator
- Eggleton, Roger B.; Kimberley, Jason S.; MacDougall, James A.
- Relation
- Journal of Combinatorial Mathematics and Combinatorial Computing Vol. 106, Issue August 2018, p. 185-207
- Relation
- https://mathscinet.ams.org/mathscinet/search/publications.html?pg1=ISSI&s1=364632
- Publisher
- Charles Babbage Research Centre
- Resource Type
- journal article
- Date
- 2018
- Description
- For any positive integer n, there are integers a, b such that a2 is the largest square divisor of n, and n = a2b: the square-free rank of n is the number of prime divisors of 6. For instance, 242 = 112.2 and 243 = 92.3 lie in a run of 5 consecutive integers of square-free rank 1. Below 1010, the longest run of integers with constant square-free rank has 20 members, all of square-free rank 3. For each k ≥ 1, we give an upper bound on the size of runs of integers of square-free rank k, and show (at least for k ≤ 7) there are infinitely many pairs of consecutive integers of square-free rank k. For distinct primes p, q there is a smallest positive integer d(p, q) such that 0 < a2p-b2q = d(p, q) ≤ p holds for infinitely many integers a, b. We study sequences (a1,a2,... ,ak) for rank 1 integers 2a2 1 > 3a2 2 > · · · > pka2 k with pi(ai + 1)2 > 2a21for 1 < i < k, and the corresponding sequences in which all the inequalities are reversed and {equation presented}. We show that (3n2 | n ∈ N) has no member in the interval [2a2,2(a+ 1)2] with a ∈ N if and only if a is in the Beatty sequence ([n (3 + √6)] | n ∈ N); several generalisations are studied. Finally, we study closest linear combinations {equation presented}.
- Subject
- combinatorial mathematics; consecutive integers; linear combinations; longest run
- Identifier
- http://hdl.handle.net/1959.13/1441260
- Identifier
- uon:41366
- Identifier
- ISSN:0835-3026
- Language
- eng
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