Puzzle 1283 A kind of flip side of Puzzle 1282

Puzzle 1283 A kind of flip side of Puzzle 1282

On September 25, 2026, Tesfamichael B. Bogale sent the following puzzle:

Let p be a prime. Build a triangle of primes as follows:

Row 1 contains just p.
Row k (k = 2, 3, 4, ..., n) contains the next (2k - 1) consecutive primes, carrying on from
wherever the previous row left off.

So the row sizes are 1, 3, 5, 7, 9, ..., n. After n rows, exactly n^2 primes have been used in total.

For each n value, it is required that the sum of the primes in each row, 2, 3, ...,n is a composite value (the
first row is always prime trivially). The triangle ends when the sum of the primes in the row n+1 is a prime.

Examples:

Example 1 (p = 2):
Row 1:                     2 (sum = 2 prime, trivial)
Row 2:                 3, 5, 7 (sum = 15 composite)
Row 3:           11, 13, 17, 19, 23 (sum = 83 prime)
So, p = 2 has one streak.

Example 2 (p = 11):
Row 1:                     11 (sum = 3 prime, trivial)
Row 2:                 13, 17, 19 (sum = 49 composite)
Row 3:           23, 29, 31, 37, 41 (sum = 161 composite)
Row 4:      43, 47, 53, 59, 61, 67, 71 (sum = 401 prime)
So, p = 11 has two streaks.

Example 3 (p = 43):
Row 1:                                    43 (sum = 3 prime, trivial)
Row 2:                              47, 53, 59 (sum = 159 composite)
Row 3:                         61, 67, 71, 73, 79 (sum = 351 composite)
Row 4:                    83, 89, 97, 101, 103, 107, 109 (sum = 689 composite)
Row 4:             113, 127, 131, 137, 139, 149, 151, 157, 163 (sum = 1267 composite)
Row 4:      167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227 (sum = 2141 prime)
So, p = 43 has four streaks.

The primes are picked based on the first smallest prime, Sp, that breaks the record held by the previous
smallest prime. In the examples above, 11 breaks the record held by 2, and 43 breaks the record held by
11. There is no prime between 2 and 11 that breaks the record held by 2. Similarly, there is no prime
between 11 and 43 that breaks the record held by 11. The process follows this trend.

I have found the smallest p, Sp, for the first fourteen n values:

n	Sp
1	2
2	11
4	43
5	71
8	179
12	223
13	467
26	607
32	3,031
59	16,447
62	94,439
82	100,829
91	972,113
104	976,909

Q1. Can you verify the values of the Table?

Q2. Can you extend this Table, and find the next (n, Sp)?