×

2 and 9 are the only biunitary superperfect numbers. (English) Zbl 1413.11011

A positive divisor \(d\) of a positive integer \(N\) is called a unitary divisor if the greatest common divisor of \(d\) and \(N/d\) equals \(1\), and a positive divisor \(d\) of \(N\) is called a biunitary divisor if the greatest common unitary divisor of \(d\) and \(N/d\) equals \(1\). Let \(\sigma^{\ast\ast}(N)\) denote the sum of the biunitary divisors of \(N\). A positive integer \(N\) is said to be biunitary perfect if \(\sigma^{\ast\ast}(N)=2N\). C. R. Wall [Proc. Am. Math. Soc. 33, 39–42 (1972; Zbl 0214.06404)] proved (applying elementary methods) that \(6\), \(60\) and \(90\) are the only biunitary perfect numbers. A positive integer \(N\) is said to be biunitary superperfect if \(\sigma^{\ast\ast}(\sigma^{\ast\ast}(N))=2N\). The present author shows (applying also only elementary methods) that \(2\) and \(9\) are the only biunitary superperfect numbers.

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas

Citations:

Zbl 0214.06404
PDFBibTeX XMLCite
Full Text: arXiv