Zeisel Numbers

Digalo con numeros
Say it with Numbers

Investigating the properties of the number associated with my diurnal age yesterday (27559), I noticed that Numbers Aplenty mentions that this number is a Zeisel number with parameters (4, 3). Now this was a type of number that I'd not heard of before. Clicking on the link in previously mentioned resource, I was given this explanation:

Let us define a sequence as:$$\begin{array}{l} p_0 = 1\\ p_n = a\cdot p_{n-1}+b\ \end{array}$$where \(a,b\in\mathbb{Z}\). If the numbers \(p_1,p_2,\dots,p_k\) are all distinct primes and \(k\ge 3\), then their product is a Zeisel number.

So applied to 27599, we start with 1 to form a new number thus \(4 \times 1 + 3 = 7\). This is prime so it is used as our new input to form the next number which is \(4 \times 7 + 3 = 31\). This is also prime so we proceed using 31 as the new input. This generates \(4 \times 31 + 3 = 127\) which is prime and this sequence of prime numbers (\(7, 31,127\)) can thus form a Zeisel number: $$7 \times 31 \times 127 = 27559$$The input 127 does not produce a prime number and so no further Zeisel numbers can be formed using (4, 3) as parameters. Interestingly, the primes 7, 31 and 127 are also consecutive Mersenne primes.

The example is given in the link of \(1419 = 3 \times 11 \times 43\) that is a Zeisel number with parameters of \(a=4\) and \(b=-1\), because \(3 = 4 \times 1-1\), \(11=4 \times 3-1\) and \(43=4 \times 11-1\).

The smallest Zeisel numbers which are the product of 3, 4, 5 and 6 factors are shown in Table 1.

Table 1: link

The first Zeisel numbers are 105, 1419, 1729, 1885, 4505, 5719, 15387, 24211, 25085, 27559, 31929, 54205, 59081, 114985, 207177 (OEIS A051015). Clearly they are few and far between as the previous such number in my diurnal age count was 25085. This was before I began keeping records in my first AirTable database. After 27599, the next is 31929, a long way off.