We will call a program that uses spigot algorithm to compute a number a spigot program. As pointed out below, the following is not one in the computational sense. But I considered using cloud functions do so, as “generating digits selectively,”
According to the Wolfram Notebook Assistant,
A spigot algorithm is a type of algorithm used to generate the digits
of a number sequentially, one at a time, in a “streaming” fashion.
This kind of algorithm is particularly useful for generating digits of
irrational numbers, such as [Pi] (pi) or e, where the digits can be
produced indefinitely without needing to compute the entire number
first.The term “spigot” is analogous to a faucet or tap, where the digits of
the number “drip out” one by one. Spigot algorithms are designed to be
efficient in terms of both time and space, often requiring only a
fixed amount of memory to produce each digit.One of the key characteristics of spigot algorithms is that they are
designed to generate each digit in a fixed amount of time, regardless
of how many digits have already been produced. This makes them
particularly useful for applications where you need a large number of
digits but don’t want to store the entire number in memory.Examples of spigot algorithms include those used for generating the
digits of [Pi], such as the Bailey[Dash]Borwein[Dash]Plouffe (BBP)
formula, which allows the calculation of the nth digit of [Pi] in
base 16 without needing to calculate the preceding digits.
Here is one that computes and verifies up to the 10,000th digit, of your inputting, of the MRB constant. It takes great advantage of the Wolfram library.
Here I inputted 2048:
Can you write one for another number?