Numerical and Statistical Analysis of Aliquot Sequences

ABSTRACT

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 ⩽ 2³⁷ and k = 1, …, 10. Second, we extend the computation of numbers not in the range of s(n) (called untouchable) by [Pollack and Pomerance 16] to the bound of 2⁴⁰ 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 76], we estimate the growth rate of terms in aliquot sequences using a Markov chain model based on data extracted from thousands of sequences.

KEYWORDS:

• aliquot sequence
• Guy–Selfridge conjecture
• Markov chain

2000 AMS SUBJECT CLASSIFICATION:

• Primary 11Y55
• Secondary 11B83,11A25

Acknowledgments

The authors wish to thank Carl Pomerance and the anonymous referee for many helpful comments and suggestions.

Notes

1 We have $\#s^{-1}\left(690100611194\right)=139$, and this is the maximum over all $\#s^{-1}\left(n\right)$ with even n ⩽ 2⁴⁰. For more details, see the OEIS sequence A283157 [Sloane 17].

Additional information

Funding

The third author's research is supported by NSERC Discovery Grant RGPIN-2016-04545.