More On Sums of Digits

Not long ago, in December of 2024, I created a post titled Prime Sums Of Digits, Digits Squared And Digits Cubed in which I looked at numbers that have a sum of digits, a sum of digits squared and a sum of digits cubed that are all prime. There are 1985 such numbers in the range up to 40000 and 322 of them are prime themselves.

In this post I want to look at numbers that not only meet the just mentioned criteria but have the additional property that, when these sums are added to the original number, the result is also prime. Here are the criteria that such numbers must meet with $\text{SOD}$ standing for Sum Of Digits:

• $\text{SOD}$ of number is prime
• ${\text{SOD}}^2$ of number is prime
• ${\text{SOD}}^3$ of number is prime
• number + $\text{SOD}$ is prime
• number + ${\text{SOD}}^2$ is prime
• number + ${\text{SOD}}^3$ is prime

As it turns out, there are only 19 such numbers in the range up to 40000 and only the first two of them are prime (permalink): 

11, 101, 166, 4490, 4528, 4630, 6016, 8254, 8788, 10066, 12422, 13166, 18284, 18688, 20854, 25570, 31166, 32518, 36064

The first composite number is 166 so let's check that it satisfies the criteria:

• $\text{SOD}$ is prime: 13
• ${\text{SOD}}^2$ is prime: 73
• ${\text{SOD}}^3$ is prime: 433
• 166 + 13 = 179 is prime
• 166 + 73 = 239 is prime
• 166 + 433 = 599 is prime

Interestingly, six of the above numbers survive the addition of a further criterion, namely that:

•  ${\text{SOD}}^4$ is prime
• number + ${\text{SOD}}^4$ is prime

These numbers are 11, 101, 4528, 6016, 10066, 20854 (permalink) and we have to look at much larger numbers to find any that satisfy yet another criterion, namely:

•  ${\text{SOD}}^5$ is prime
• number + ${\text{SOD}}^5$ is prime

The only two numbers that satisfy this additional criteria are 100001 and 104930. The former number is not prime but composite since 100001 = 11 x 9091. It would be interesting to see how much further this process could be taken but we would then be looking at very large numbers indeed.