OFFSET
1,1
COMMENTS
Except for the prime sum of the single number 2 in the first row, each sum is equal to a twin prime.
For row k, the k! sums produce as many distinct primes, or, for row k >= 2, k!/2 distinct twin prime pairs.
Defining this same triangle from another angle, e.g., by discovering and describing a regularity in its structure, and/or proving its infinitude, is equivalent to proving the twin prime conjecture. An independent proof of the latter may not prove the infiniteness of this sequence, however.
EXAMPLE
Triangle T(n/k) begins:
n/k | 1 2 3 4
--------------------------------------------
1 | 2;
2 | 3, 5;
3 | 6, 12, 24;
4 | 18030, 97830, 165690, 392250;
...
a(5) = 12 because in row three, after a(4), a(5) is the second of the three least values in the row: both a(1) + a(2) + a(5) = 2 + 3 + 12 = 17 and a(1) + a(3) + a(5) = 2 + 5 + 12 = 19 are primes.
Below are the prime factors of the terms. These are listed in order of magnitude and without exponents or multiplication symbols, for clarity:
2
3 5
2 3 2 2 3 2 2 2 3
2 3 5 601 2 3 3 5 1087 2 3 3 5 7 263 2 3 5 5 5 523
CROSSREFS
KEYWORD
AUTHOR
Tamas Sandor Nagy, Mar 14 2023
EXTENSIONS
a(7)-a(10) from Thomas Scheuerle, Mar 14 2023
STATUS
approved