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%I #55 Nov 06 2023 07:16:24
%S 2,3,5,6,12,24,18030,97830,165690,392250
%N 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.
%C Except for the prime sum of the single number 2 in the first row, each sum is equal to a twin prime.
%C For row k, the k! sums produce as many distinct primes, or, for row k >= 2, k!/2 distinct twin prime pairs.
%C 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.
%e Triangle T(n/k) begins:
%e n/k | 1 2 3 4
%e --------------------------------------------
%e 1 | 2;
%e 2 | 3, 5;
%e 3 | 6, 12, 24;
%e 4 | 18030, 97830, 165690, 392250;
%e ...
%e 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.
%e Below are the prime factors of the terms. These are listed in order of magnitude and without exponents or multiplication symbols, for clarity:
%e 2
%e 3 5
%e 2 3 2 2 3 2 2 2 3
%e 2 3 5 601 2 3 3 5 1087 2 3 3 5 7 263 2 3 5 5 5 523
%Y Cf. A000040, A001097, A000217, A000142.
%K nonn,tabl,more,hard
%O 1,1
%A _Tamas Sandor Nagy_, Mar 14 2023
%E a(7)-a(10) from _Thomas Scheuerle_, Mar 14 2023
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