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A variation on perfect numbers. (English) Zbl 1086.11007

Define \(s_k(n)= \sum p_{i_1} p_{i_2}\cdots p_{i_k}\), where the sum is taken over all products of \(k\) prime factors of \(n= p_1p_2\cdots p_r\) (\(p_i\) primes, not necessary different). The author calls a natural number \(n\) \(k\)-symmetric-perfect (\(k\)-SP), if \(s_k(n)= n\) and \(\Omega(n)> k\). For \(k= 1,2,3\) he finds all \(k\)-SP numbers (4 resp. 27, 48 resp. none). For every \(k\in\mathbb{N}\) there are only finitely many \(k\)-SP numbers. If \(k= 1\) or \(2\), every large number has the form \(s_k(n)\).

MSC:

11A25 Arithmetic functions; related numbers; inversion formulas
11Y70 Values of arithmetic functions; tables
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Online Encyclopedia of Integer Sequences:

Values of r(k): the smallest r such that binomial(r, k) < 2^{r-k}.