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.

2, 3, 5, 6, 12, 24, 18030, 97830, 165690, 392250

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

Table of n, a(n) for n=1..10.

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

Cf. A000040, A001097, A000217, A000142.

Sequence in context: A007435 A125877 A118787 * A191783 A358290 A098930

Adjacent sequences: A359665 A359666 A359667 * A359669 A359670 A359671

Keyword

nonn,tabl,more,hard

Author

Tamas Sandor Nagy, Mar 14 2023

Extensions

a(7)-a(10) from Thomas Scheuerle, Mar 14 2023

Status

approved