Subject: Analysis and Computation of a Unique Number Sequence
Date: June 10, 2023
This memo is to document the findings from the analysis of a specific number sequence. The sequence begins with 2, 4, and 8 and then employs a unique iterative process for generating subsequent numbers.
This generation process includes identifying the last number, or ‘base,’ added to the sequence. This base is then added to every second number in the sequence that is smaller than the base. The sums are appended to the sequence, and the base is then doubled and added to the sequence. This process repeats until the sequence reaches a desired length.
The objective was to identify the largest number in the sequence that is less than 2^64. After an extensive computation process, which included translating C functions into PARI/GP code for handling large integers, the number 18446462598733454336 was identified.
This number, 18446462598733454336, is the largest number in the sequence that falls below 2^64. It should be noted that there are approximately 3000 other numbers in the sequence between this number and 2^64, as dictated by the sequence generation rules.
This exploration required the utilization of arbitrary-precision arithmetic tools, such as PARI/GP, due to the size of the numbers involved. The investigation provides key insights into the properties of such mathematically generated sequences and demonstrates the application of programming and computational tools in solving complex mathematical problems.
This memo serves to document the analytical process and the key findings of this exploration.
End of memo.
