1-2-3-4 Factor Chains And Beyond

The number associated with my diurnal age today, 27271, has an interesting property. Firstly, it's prime. Now if we multiply it by 2 and add 1, we don't get another prime (which would make 27271 a Sophie Germain prime) but instead we get a semiprime:$$ \begin{align} 2 \times 27271+1 &= 54543 \\ &= 3 \times 18181 \end{align}$$Now if we repeat this process with the newly generated semiprime, we get a number with three factors:$$ \begin{align} 2 \times 54543+1 &= 109087 \\ &= 11 \times 47 \times 211 \end{align}$$Now repeating the process again, we end up with a number that has four (not distinct) prime factors:$$ \begin{align} 2 \times 109087+1 &= 218175 \\ &= 3 \times 5^2 \times 2909 \end{align} $$Numbers with this property form OEIS A235646 (permalink):

A235646

Primes \(p\) such that \(b=2 \times p+1\) is semiprime, \(c=2 \times b+1\) is 3-almost prime and \(d=2 \times c+1\) is 4-almost prime.

The initial members of the sequence are:

43, 1429, 2239, 3319, 4831, 6379, 8821, 10501, 11383, 12781, 13003, 14771, 15091, 16063, 16759, 18223, 19213, 19681, 20021, 22571, 24103, 24109, 24571, 25939, 27271, 28933, 29833, 30241, 31723, 33679, 33811, 34381, 34781, 35591, 35863, 39373

As can be seen, such primes are not frequent and it will be quite some time before I celebrate another prime with this property. In fact, it won't occur until Tuesday, June 20th 2028.

The natural question to ask is can we extend this chain further so we have a 1-2-3-4-5 chain. The answer is yes. In the range up to one million, the primes with this property are:

197161, 267341, 283181, 470863, 543463, 646423, 751759, 911321, 934981

Let's take the first number in this sequence, 197161: $$ \begin{align} 2 \times 197161 +1 &= 394323\\ &= 3 \times 131441\\ 2 \times 394323 +1 &= 788647\\ &= 17 \times 23 \times 2017 \\2 \times 788647 + 1  &= 1577295\\ &= 3^2 \times 5 \times 35051\\2 \times 1577295 + 1&= 3154591 \\ &=11^2 \times 29^2 \times 31 \end{align}$$Can we go further? Well, in the range up to ten million, there are only two primes with this 1-2-3-4-5-6 property and they are 8651161 and 9723331 (permalink). Here are the factorisations for both numbers (permalink):

8651161

17302323 = 3 * 5767441

34604647 = 7 * 11 * 449411

69209295 = 3 * 5 * 17 * 271409

138418591 = 19^2 * 37 * 43 * 241

276837183 = 3^4 * 7 * 488249

9723331

19446663 = 3 * 6482221

38893327 = 11 * 37 * 95561

77786655 = 3 * 5 * 331 * 15667

155573311 = 19^2 * 23 * 41 * 457

311146623 = 3^3 * 53 * 103 * 2111

Of course, multiplication by 2 could be changed to multiplication by 3 (shades of Collatz) and in the range up to ten million there is only one prime that follows a 1-2-3-4-5-6 progression and that is 9203191 with the following progression:

9203191

27609574 = 2 * 13804787

82828723 = 283 * 541^2

248486170 = 2 * 5 * 59 * 421163

745458511 = 7^2 * 137 * 293 * 379

2236375534 = 2 * 17 * 37 * 61 * 151 * 193

A multiplication factor of 4 yields the primes 1876711, 6840241, 7704877, 9369589 (permalink) in the range up to ten million. Looking at the first prime 1876711 we get the following progression (permalink):

1876711

7506845 = 5 * 1501369

30027381 = 3 * 251 * 39877

120109525 = 5^2 * 401 * 11981

480438101 = 11 * 13^2 * 109 * 2371

1921752405 = 3^3 * 5 * 509 * 27967

I could keep going that that's probably enough. However, what if we increased the multiplication factor by 1 each time so that we have 2, 3, 4, 5, 6 etc. Well, in the range up to ten million, there is only one prime that satisfies the 1-2-3-4-5-6 progression and that is 2857427 (permalink) with the progression (permalink):

2857427

5714855 = 5 * 1142971

17144566 = 2 * 47 * 182389

68578265 = 5 * 7 * 859 * 2281

342891326 = 2 * 17 * 103 * 179 * 547

2057347957 = 7^3 * 29 * 107 * 1933

Of course if we specified that the prime factors must be distinct in the previous analyses, the previously discussed sequences would be drastically culled but that's perhaps a topic for a future post. Overall, let's not forget that all the properties just discussed are independent of the number base that is used.