We present a variety of numerical data related to the growth of terms in aliquot sequences, iterations of the function s(n) = σ(n) − n. First, we compute the geometric mean of the ratio s k (n)/s k−1 (n) of kth iterates for n ≤ 2 37 and k = 1, . . . , 10. Second, we extend the computation of numbers not in the range of s(n) (called untouchable) by Pollack and Pomerance [2016] to the bound of 2 40 and use these data to compute the geometric mean of the ratio of consecutive terms limited to terms in the range of s(n). Third, we give an algorithm to compute k-untouchable numbers (k − 1st iterates of s(n) but not kth iterates) along with some numerical data. Finally, inspired by earlier work of Devitt [1976], we estimate the growth rate of terms in aliquot sequences using a Markov chain model based on data extracted from thousands of sequences.
We present an improved algorithm for tabulating class groups of imaginary quadratic fields of bounded discriminant. Our method uses classical class number formulas involving theta-series to compute the group orders unconditionally for all ∆ ≡ 1 (mod 8). The group structure is resolved using the factorization of the group order. The 1 mod 8 case was handled using the methods of [JRW06], including the batch verification method based on the Eichler-Selberg trace formula to remove dependence on the Extended Riemann Hypothesis. Our new method enabled us to extend the previous bound of |∆| < 2 · 10 11 to 2 40 . Statistical data in support of a variety conjectures is presented, along with new examples of class groups with exotic structures.
Let F (x, y) be a binary form of degree at least three and non-zero discriminant. In this article we compute the automorphism group Aut F for four families of binary forms. The first two families that we are interested in are homogenizations of minimal polynomials of 2 cos 2π n and 2 sin 2π n , which we denote by Ψn(x, y) and Πn(x, y), respectively. The remaining two forms that we consider are homogenizations of Chebyshev polynomials of first and second kinds, denoted Tn(x, y) and Un(x, y), respectively.
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