# [seqfan] A classification of the positive integers over primes

Wed Sep 17 09:50:42 CEST 2014

```I would like to propose a rather natural classification of positive numbers over their specially defined proximity to primes. Essentially, it details classification {{1}, prime numbers, composite numbers} among many others known classifications.
Firstly, consider an importat M. J. Hardy's comment in A156759: "Except for a(1)=2, this is the sequence of numbers k such that the smallest prime factor of k is the largest prime less than or equal to the square root of k". Proof of this statement is evident: it is sufficient to note that the terms (except for a(1)) enlarge their smallest prime divisor only on prime(n)^2. Due to this comment, I call terms of A156759 beginning with the second one "preprimes". Indeed, for a term N, using the standard primality test with division N by consecutive primes <= sqrt(N), we only on the last step conclude that N is not prime. In connection with this, I consider a classification of the positive numbers with classes {1}, A000040(primes), A156759(n>=2)(preprimes, or preprimes of the first kind), A247393 (preprimes of the second kind), A247394 (preprimes of the third kind), etc., where n is called a preprime of the m-th kind, if its least prime divisor is the m-th maximal prime <= sqrt(n) (we say that an increasing finite sequence a_1<a_2<...<a_h has the first maximum a_h, the second maximum a_(h-1), etc.)
The first numbers of such classes are 1, 2, 4, 10, 26, 50,...( A247395) . Note that, every class contains only a finite numbers with a given least prime divisor. For example, numbers of even numbers over the classes are 0, 1, 3, 8, 12,... .(cf. A247396; it is easy to see that, for n>=3, the terms here are (prime(n)^2-prime(n-1)^2)/2).
Many questions arise concerning this classification. I thank Peter for extension of sequences A247393-A247395. I wonder have anyone already met a similar construction?

Best regards,