Meertens number

Definition

Let n be a natural number. We define the Meertens function for base b 1 F_{b} : \mathbb{N} \rightarrow \mathbb{N} to be the following: :F_{b}(n) = \prod_{i=0}^{k - 1} p_{k - i - 1}^{d_i}. where k = \lfloor \log_{b}{n} \rfloor + 1 is the number of digits in the number in base b, p_i is the i-th prime number (starting at 0), and :d_i = \frac{n \bmod{b^{i+1}} - n \bmod b^i}{b^i} is the value of each digit of the number. A natural number n is a Meertens number if it is a fixed point for F_{b}, which occurs if F_{b}(n) = n. This corresponds to a Gödel encoding.

For example, the number 3020 in base b = 4 is a Meertens number, because :3020 = 2^{3}3^{0}5^{2}7^{0}.

A natural number n is a sociable Meertens number if it is a periodic point for F_{b}, where F_{b}^k(n) = n for a positive integer k, and forms a cycle of period k. A Meertens number is a sociable Meertens number with k = 1, and a amicable Meertens number is a sociable Meertens number with k = 2.

The number of iterations i needed for F_{b}^{i}(n) to reach a fixed point is the Meertens function's persistence of n, and undefined if it never reaches a fixed point.

Meertens numbers and cycles of ''F_b'' for specific ''b''

All numbers are in base b.

References

References

1. [[Richard S. Bird]]. (1998). "Meertens number". [[Journal of Functional Programming]].
2. {{OEIS