1-2-3 Primes and Beyond

What do I mean by a 1-2-3 prime? Well, let's use 13 and an example because this is the first 1-2-3 prime. The name is given because this prime and the subsequent two numbers can be written in a special way, namely:$$ \begin{align} 13&= 1 \times 13\\14& = 2 \times 7\\15&=3 \times 5 \end{align}$$The 1-2-3 primes constitute OEIS A163573:

A036570

Primes \(p\) such that \( \dfrac{p+1}{2} \) and \( \dfrac{ p+2}{3} \) are also primes.

The initial members are:

13, 37, 157, 541, 877, 1201, 1381, 1621, 2017, 2557, 2857, 3061, 4357, 4441, 5077, 5581, 5701, 6337, 6637, 6661, 6997, 7417, 8221, 9181, 9661, 9901, 10837, 11497, 12457, 12601, 12721, 12757, 13681, 14437, 15241, 16921, 17077, 18217

Looking at the second member of the sequence, 37, it can seen that: $$ \begin{align} 37&= 1 \times 37\\38& = 2 \times 19\\39&=3 \times 13 \end{align}$$By a 1-2-3-4 prime, I mean a prime such as 12721 where this prime and the subsequent three numbers can be written in a special way, namely:$$ \begin{align} 12721&= 1 \times 12721\\12722& = 2 \times 6361\\12723&=3 \times 4261\\ 12724 &=4 \times 3181 \end{align}$$The 1-2-3-4 primes constitute OEIS A036570:

A163573

Primes \( p \) such that \( \dfrac{p+1}{2} \),  \( \dfrac{p+2}{3} \) and  \( \dfrac{p+3}{4} \) are also primes.

The initial members of the sequence are:

12721, 16921, 19441, 24481, 49681, 61561, 104161, 229321, 255361, 259681, 266401, 291721, 298201, 311041, 331921, 419401, 423481, 436801, 446881, 471241, 525241, 532801, 539401, 581521, 600601, 663601, 704161, 709921, 783721, 867001, 904801

Looking at the second member of the sequence, 16921, it can be seen that:$$ \begin{align} 16921&= 1 \times 16921\\16922& = 2 \times 8461\\16923&=3 \times 5641\\ 16921 &=4 \times 4231 \end{align}$$The comments to the second OEIS sequence state that all terms are of the form \(120k+1 \), which is why they all end 1. This makes it easy to find the increasingly rare 1-2-3-4-5 and 1-2-3-4-5-6 primes because the sequence will be:$$\begin{align} \frac{120k+1}{1}&=120k+1\\ \frac{120k+2}{2}&=60k+1\\ \frac{120k+3}{3}&=40k+1\\ \frac{120k+4}{4}&=30k+1\\ \frac{120k+5}{5}&=24k+1\\ \frac{120k+6}{6}&=20k+1 \end{align}$$All we need so then is to check each of the \(k\) expressions for primeness. So, armed with this knowledge, let's look for 1-2-3-4-5 and 1-2-3-4-5-6 primes. Starting with the former, we find the following after testing for values of  \(k\) up to 10,000:

• [19441, 9721, 6481, 4861, 3889]
• [266401, 133201, 88801, 66601, 53281]
• [423481, 211741, 141161, 105871, 84697]
• [539401, 269701, 179801, 134851, 107881]
• [600601, 300301, 200201, 150151, 120121]
• [663601, 331801, 221201, 165901, 132721]
• [908041, 454021, 302681, 227011, 181609]
• [1113961, 556981, 371321, 278491, 222793]

The initial 1-2-3-4-5 primes are:

[19441, 266401, 423481, 539401, 600601, 663601, 908041, 1113961]

These terms form part of OEIS A204592:

A204592

Primes \(p\) such that \( \dfrac{p+1}{2} \), \( \dfrac{p+2}{3} \), \( \dfrac{p+3}{4} \) and \( \dfrac{p+4}{5} \) are also prime.

Here are more terms:

19441, 266401, 423481, 539401, 600601, 663601, 908041, 1113961, 1338241, 1483561, 1657441, 1673401, 2578801, 3109681, 3150841, 3336601, 3613681, 4112761, 4160641, 4798081, 5114881, 5412961, 5516281, 5590201, 5839681, 6078361, 7660801, 8628481, 9362641, 9388801, 9584401, 9733081

To find the initial 1-2-3-4-5-6 primes, the value of \(k\) must be set much higher. Here is what we find for values of \(k\) up 1,000,000:

• [5516281, 2758141, 1838761, 1379071, 1103257, 919381]
• [16831081, 8415541, 5610361, 4207771, 3366217, 2805181]
• [18164161, 9082081, 6054721, 4541041, 3632833, 3027361]
• [29743561, 14871781, 9914521, 7435891, 5948713, 4957261]
• [51755761, 25877881, 17251921, 12938941, 10351153, 8625961]

The initial 1-2-3-4-5-6 primes are:

[5516281, 16831081, 18164161, 29743561, 51755761]

These terms form part of OEIS

A208455

Primes \(p\) such that \( \dfrac{p+k}{k+1}\) is a prime number for \(k=1, \dots,5\).

Here are some more terms in the sequence:

5516281, 16831081, 18164161, 29743561, 51755761, 148057561, 153742681, 158918761, 175472641, 189614881, 212808961, 297279361, 298965241, 322030801, 467313841, 527428441, 661686481, 668745001, 751524481, 808214401

Here is a SageMathCell permalink that can be used to determine these primes.