Meertens number

Let $n$ be a natural number. We define the Meertens function for base $b\; >\; 1$ $F\_\{b\}\; :\; \backslash mathbb\{N\}\; \backslash rightarrow\; \backslash mathbb\{N\}$ to be the following:

:$F\_\{b\}(n)\; =\; \backslash prod\_\{i=0\}^\{k\; -\; 1\}\; p\_\{k\; -\; i\; -\; 1\}^\{d\_i\}.$

where $k\; =\; \backslash lfloor\; \backslash log\_\{b\}\{n\}\; \backslash rfloor\; +\; 1$ is the number of digits in the number in base $b$, $p\_i$ is the $i$-th prime number (starting at 0), and

:$d\_i\; =\; \backslash frac\{n\; \backslash bmod\{b^\{i+1\}\}\; -\; n\; \backslash bmod\; b^i\}\{b^i\}$

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.

All numbers are in base $b$.

• Arithmetic dynamics
• Dudeney number
• Factorion
• Happy number
• Kaprekar's constant
• Kaprekar number
• Narcissistic number
• Perfect digit-to-digit invariant
• Perfect digital invariant
• Sum-product number

• {{OEIS el|1=A189398|2=a(n) = 2^d(1) * 3^d(2) * ... * prime(k)^d(k)|formalname=a(n) = 2^d(1) * 3^d(2) * ... * prime(k)^d(k), where d(1)d(2)...d(k) is the decimal representation of n}}
• {{OEIS el|sequencenumber=A246532|name=Smallest Meertens number in base n, or -1 if none exists.}}

{{Classes of natural numbers}}

Category:Arithmetic dynamics

Category:Base-dependent integer sequences