Puzzle 1256 Another sequence of primes

 On Jan 6, 2026, Palolo sent this nice puzzle.

Consider the following process: take a prime k>9, sum the digits, repeat the sum deleting the first addendum and adding the previous sum and so on. 

Sequence lists the minimum prime k that produces a run of exactly n consecutive primes. (this sequence is not yet present in OEIS)

I've found the first 10 terms:

0   [13]

1   [23, 5]

2   [101, 2, 3]

3   [11, 2, 3, 5]

4   [317, 11, 19, 37, 67]

5   [331, 7, 11, 19, 37, 67]

6   [599, 23, 41, 73, 137, 251, 461]

7   [35311, 13, 23, 41, 79, 157, 313, 613]

8   [3393311, 23, 43, 83, 157, 311, 619, 1237, 2473]

9   [3533377, 31, 59, 113, 223, 443, 883, 1759, 3511, 6991]

10   [1933537, 31, 61, 113, 223, 443, 881, 1759, 3511, 6991, 13921]

11   [331733953, 37, 71, 139, 277, 547, 1091, 2179, 4349, 8693, 17383, 34729]

 

Example: k(5) = 331 because:

1) 3 + 3 + 1 = 7, prime;
2) 3 + 1  + 7 = 11, prime;
3) 1 + 7 + 11 = 19, prime;
4) 7 + 11 + 19 = 37, prime;
5) 11 + 19 + 37 = 67, prime, and 19 + 37 + 67 = 123 composite.

 

Q. Can you extend this sequence?

Last hour news: Less than a couple of hours after publishing this puzzle I received an email from Michael Branicky, who wrote:

"This is now in the OEIS as https://oeis.org/A391445 .