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%I #16 Dec 02 2019 13:55:34
%S 2,3,2,4,3,2,5,5,3,2,6,7,5,3,2,7,8,7,5,3,2,8,9,8,7,5,3,2,9,11,9,11,7,
%T 5,3,2,10,13,11,13,11,7,5,3,2,11,14,13,16,13,11,7,5,3,2,12,15,14,17,
%U 16,13,11,7,5,3,2,13,17,15,19,17,17,13,11
%N Table of A(n,k) read by antidiagonals, where A(n,1)=2, and every n+1 consecutive terms in row n are pairwise coprime. Terms are chosen to be the least increasing value compatible with these constraints.
%C This algorithm acts as a prime number sieve. Prime numbers move to the left with each step. The second diagonal (and all the numbers to the left) are all primes.
%C The first composite number in each row: 4, 8, 8, 16, 16, 24, 24, 32, 32, 32, 45, 48, 48, 54, 64, 64, 64, 72, 80, 81, 90, 96, 105, 108, 108, 120, 128, 128, 128, ....
%C In this sieve, some numbers disappear and then reappear. For example, 26 disappears on the third row, then reappears on the 4th and 5th rows, then disappears again.
%e Table begins:
%e 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
%e 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 25, ...
%e 2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 22, 23, 25, 27, ...
%e 2, 3, 5, 7, 11, 13, 16, 17, 19, 21, 23, 25, 26, 29, 31, 33, ...
%e 2, 3, 5, 7, 11, 13, 16, 17, 19, 21, 23, 25, 26, 29, 31, 33, ...
%e 2, 3, 5, 7, 11, 13, 17, 19, 23, 24, 25, 29, 31, 37, 41, 43, ...
%e 2, 3, 5, 7, 11, 13, 17, 19, 23, 24, 25, 29, 31, 37, 41, 43, ...
%e 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 32, 35, 37, 39, 41, ...
%e 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 32, 37, 41, 43, 45, ...
%e 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 32, 37, 41, 43, 45, ...
%e E.g., in the third row, a(3,1)=2, and every 4 consecutive terms are pairwise coprime.
%o (PARI) row(N,howmany=100)=my(v=List(primes(N))); for(i=N+1,howmany, my(L=lcm(v[#v-N+1..#v]), n=v[#v]); while(gcd(n,L)>1, n++); listput(v,n)); Vec(v) \\ _Charles R Greathouse IV_, Oct 27 2019
%Y Cf. A047255, A062062, A062063.
%K nonn,tabl
%O 1,1
%A _Ali Sada_, Oct 26 2019