Multiperfect numbers on lines of the Pascal triangle
A positive integer \(n\) is said to be \(k\)-fold perfect (or multiperfect or multiply perfect) if \(\sigma(n) = kn\). The 3-fold perfect numbers are also called triperfect, and only six of them are known: 120, 672, 523776, 459818240, 1476304896, 51001180160. Several multiperfect numbers are also known for every \(k\leq 11\). Their number varies from thousands for \(k = 8, 9, 10\), to only one for \(k = 11\), which has more than a thousand decimal digits and was discovered in 2001. Descartes discovered the first 4-fold number, and Fermat the first 5-fold number. Except for the well-known Euclid-Euler rule for \(k = 2\), no formula to generate multiperfect numbers is known. Lehmer proved that if \(n\) is odd, then \(n\) is perfect just if \(2n\) is 3-fold perfect. No odd multiperfect number is known. There are various extensions of perfect and multiperfect numbers in the literature. For example, a positive integer \(n\) is called \((m, k)\)-perfect if \(\sigma^{(m)}(n) =kn\), where \(\sigma^{(m)}\) denotes the \(m\)th fold iterate of the sum of divisors function \(\sigma\). The present authors study the multiperfect and \((2, k)\)-perfect numbers on straight lines through the Pascal triangle. They show that except for the lines parallel to the edges, all other lines through the Pascal triangle contain at most finitely many multiperfect and \((2, k)\)-perfect numbers. The authors also show, among others, that the only \(n\) such that \(B_n\) is multiperfect is \(n = 2\) and the only \(n\) such that \(C_n\) is multiperfect is \(n = 1\), where \(B_n={2n\choose n}\) is the \(n\)th middle binomial coefficient and \(C_n={1\over n+1} B_n\) is the \(n\)th Catalan number. For \((2, k)\)-perfect numbers these values are \(n = 1\) and \(n = 1, 2, 5\). Further, the authors study the distribution of the numbers \(\sigma(n)/n\) whenever the positive integer \(n\) ranges through the binomial coefficients on a fixed line through the Pascal triangle. For a general account of perfect and related numbers, see [\textit{J. Sándor} and \textit{B. Crstici}, Handbook of number theory. II. Dordrecht: Kluwer Academic Publishers (2004; Zbl 1079.11001)].
scientific article; zbMATH DE number 5554701