By MS we mean 2, 4, 8, 12, 16, 20, 28, 32, 36, 44, 52, 64, 68, 76, 84, 96, 108, 128, 132, 140,… which gets its name from the fact that the indices of the powers of two are: 1,2,3,5,8,12,… which, upon adding 1, gives 2, 3, 4, 6, 9, 13, … a sequence recording the growth of a population of cows in the game Minecraft, i.e. c(1)=2, c(n)=floor(3a(n-1)/2) ).
This was collaborative work with ChatGPT4, which below provides a summary of our exchanges or discussions.
Over our recent discussions, we delved into various properties and conjectures associated with the Minecraft Sequence (MS). This sequence, originating from a popular video game, exhibits intriguing mathematical behavior. Notably, we investigated the distribution of the sequence around powers of two and developed an effective numerical method to locate the sequence values around 2^K.
Our findings indicate an interesting regularity: the ‘gap’ between 2^K and the nearest term in the sequence seems to follow a power-law relationship with K, specifically gap(2^K) ≈ (2^K)^0.63. This characteristic was consistently observed across various powers of two, and the origin of this exponent (0.63) remains a mystery.
Additionally, we discussed potential limit behavior of the MS. By considering the logarithmic transformation of the sequence, we noted that the behavior closely resembles that of the Blancmange function, a continuous but nowhere differentiable function. This resemblance suggests that the MS may exhibit fractal-like properties, displaying complexity at every scale.
Furthermore, the calculate_index function was developed and proved extremely useful in our investigations. However, its computational limitations prompted us to consider other algorithmic improvements.
Moving forward, suggested research avenues include a deeper exploration of the limit behavior of the sequence, refining the calculate_index function, and an analysis of the properties of the gap between sequence terms around powers of two. This could lead to a deeper understanding of the mathematical principles underlying the Minecraft Sequence and might also shed light on its mysterious connections with other areas of mathematics.
