A328739 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.

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, 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, 16, 13, 11, 7, 5, 3, 2, 13, 17, 15, 19, 17, 17, 13, 11

Offset: 1

Author

Ali Sada, Oct 26 2019

Keywords

• nonn
• tabl

Comments

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.

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, ....

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.

Examples

			Table begins:
  2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, ...
  2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 20, 21, 23, 25, ...
  2, 3, 5, 7, 8, 9, 11, 13, 14, 15, 17, 19, 22, 23, 25, 27, ...
  2, 3, 5, 7, 11, 13, 16, 17, 19, 21, 23, 25, 26, 29, 31, 33, ...
  2, 3, 5, 7, 11, 13, 16, 17, 19, 21, 23, 25, 26, 29, 31, 33, ...
  2, 3, 5, 7, 11, 13, 17, 19, 23, 24, 25, 29, 31, 37, 41, 43, ...
  2, 3, 5, 7, 11, 13, 17, 19, 23, 24, 25, 29, 31, 37, 41, 43, ...
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 32, 35, 37, 39, 41, ...
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 32, 37, 41, 43, 45, ...
  2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 32, 37, 41, 43, 45, ...
E.g., in the third row, a(3,1)=2, and every 4 consecutive terms are pairwise coprime.
		

Crossrefs

Cf. A047255, A062062, A062063.

Programs

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

Formula

A(n, k) = prime(k) if k <= n+1. - M. F. Hasler, May 09 2025