Sloane's Gap. Mathematical and Social Factors Explain the Distribution of Numbers in the OEIS

Abstract: The Online Encyclopedia of Integer Sequences (OEIS) is made up of thousands of numerical sequences considered particularly interesting by some mathematicians. The graphic representation of the frequency with which a number n as a function of n appears in that database shows that the underlying function decreases fast, and that the points are distributed in a cloud, seemingly split into two by a clear zone that will be referred to here as "Sloane's Gap". The decrease and general form are explained by mathematics, but an explanation of the gap requires further considerations.

• The paper identifies "Sloane's Gap," an unexpected separation in the distribution of number frequencies in the OEIS, which broadly follows a k/n^1.33 pattern.
• While algorithmic complexity suggests simpler numbers should appear more often, this theory alone fails to explain the observed gap in the OEIS distribution.
• The authors propose that social factors and human biases, favoring numbers like primes and powers of two, significantly influence OEIS content and likely cause Sloane's Gap.

Overview of Sloane's Gap: Exploring Mathematical and Social Influences on Integer Distribution

This paper investigates the peculiar distribution of numbers within the Online Encyclopedia of Integer Sequences (OEIS), focusing specifically on an observed phenomenon termed "Sloane's Gap." The primary objective is to analyze the significance and frequency of occurrence of integers, denoted as $N(n)$, in the OEIS. The study endeavors to understand whether mathematical factors, particularly those related to algorithmic complexity, can predict the distribution of integers or whether social factors play a role.

Key Observations

1. Distribution Characteristics: The paper presents a statistical analysis indicating that $N(n)$, the frequency of occurrences of a number $n$ in the OEIS, significantly varies but clusters around the function $\frac{k}{n^{1.33}}$ with $k \approx 2.53 \times 10^8$. Unexpectedly, this distribution forms two distinct sub-clusters separated by a gap, termed "Sloane's Gap."
2. Algorithmic Complexity Framework: The paper applies concepts from algorithmic complexity theory to hypothesize that a number's frequency $N(n)$ correlates inversely with its algorithmic complexity $K(n)$. According to this theory, the lower $K(n)$, the more 'simple' or 'interesting' the number, hence the higher the $N(n)$. However, algorithmic complexity alone does not accurately predict the observed gap in the distribution.
3. Social Influence Hypothesis: The authors propose that the gap could stem from mathematicians' biases or social preferences that influence the kinds of sequences submitted to the OEIS. This leads to certain numbers, possibly deemed more culturally or mathematically significant, appearing more frequently. For example, primes, powers of two, and numbers with many factors commonly place above the gap, suggesting human interest rather than inherent mathematical properties.

Implications and Future Directions

The findings highlight the need to consider both mathematical rigor and the broader cultural and social context in understanding mathematical phenomena. The dual influence underscores the potential for further interdisciplinary research combining algorithmic information theory and cognitive psychology to explore how subjective decision-making processes shape perceived mathematical importance.

The paper opens avenues for future research to examine:

• Extended Algorithmic Models: Developing models that incorporate more nuanced aspects of human psychology and social influences on sequence selection and recognition.
• Broader Data Analysis: Conducting longitudinal studies on the OEIS to observe changes in sequence distribution over time, potentially uncovering shifts in mathematical interest or discovery trends.
• Cognitive Impact Assessment: Investigating how awareness of shared mathematical structures influences the collective cultural understanding and representation of numbers.

In summary, while algorithmic complexity offers a valuable framework for analyzing number properties in the OEIS, the prevalence of Sloane's Gap suggests that social and cultural dimensions must also be acknowledged to fully comprehend the nuances behind the significance attributed to numerical sequences.