I would like to share with you a numerical pattern I have discovered while studying a very simple recursive algorithm.

The construction starts with:

f(1) = 1

At each step, we search for the smallest prime number p > f(n)+1 such that:

R(n) = p - f(n)

is a composite number.

Then we define:

f(n+1) = f(n) × R(n)

The first values generated by this process are:

R(n) = 4, 9, 25, 77, 323, 713, 169, 1189, 1961, 1849, ...

Their prime factorizations are:

4 = 2 × 2 = 2²

9 = 3 × 3 = 3²

25 = 5 × 5 = 5²

77 = 7 × 11

323 = 17 × 19

713 = 23 × 31

169 = 13 × 13 = 13²

1189 = 29 × 41

1961 = 37 × 53

1849 = 43 × 43 = 43²

The first remarkable observation is:

All the values R(n) found so far appear to be semiprimes, meaning the product of exactly two prime numbers.

So:

R(n) = p × q

or, in the special case where the two primes coincide:

R(n) = p²

The second observation is even more surprising:

The prime factors appearing in the different values of R(n) never seem to repeat.

For example:

77 introduces the primes 7 and 11.

323 introduces 17 and 19.

713 introduces 23 and 31.

1189 introduces 29 and 41.

Each step seems to introduce new prime "building blocks".

After checking the first 48 values:

• no prime factor appears in two different R(n);

• whenever a prime appears twice, it only happens as a perfect square:

2², 3², 5², 13², 43², 167², 241², 251², 283², 359², 419²...

Another interesting observation:

The prime factors appearing in the first 48 values cover all primes up to:

487

The first prime not appearing yet as a factor is:

491

This leads to the following conjecture:

Starting from:

f(1) = 1

and defining:

R(n) = p - f(n)

where p is the smallest prime greater than f(n)+1 for which R(n) is composite,

then:

1. Every R(n) is always a semiprime.

2. The prime factors of different R(n) values are never reused.

3. Every prime number eventually appears as a factor exactly once, except in the special case where it appears as a square: pxp=p²