

A359668


Triangle read by rows. Each term of the triangle is positive and distinct. In row k are the next k least numbers such that the sum of any one number from each of the first k rows is a prime number.


0




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.


LINKS



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


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STATUS

approved



