A362777 Triangular array read by rows: T(n,k) = n!*k + 1, n >= 1, 1 <= k <= n.

2, 3, 5, 7, 13, 19, 25, 49, 73, 97, 121, 241, 361, 481, 601, 721, 1441, 2161, 2881, 3601, 4321, 5041, 10081, 15121, 20161, 25201, 30241, 35281, 40321, 80641, 120961, 161281, 201601, 241921, 282241, 322561, 362881, 725761, 1088641, 1451521, 1814401, 2177281, 2540161, 2903041, 3265921

Offset: 1

Author

Joe B. Stephen, May 03 2023

Keywords

• tabl
• nonn

Comments

These numbers are used in a simple proof of the infinitude of the primes: n!*i + 1 and n!*j + 1 are coprime for 1 <= i < j <= n, so for any n we get n coprime integers (greater than 1) and hence we get at least n distinct primes.

Examples

			Triangle T(n,k) begins:
  n\k  1    2    3    4    5    6 ...
  1    2
  2    3    5
  3    7   13   19
  4   25   49   73   97
  5  121  241  361  481  601
  6  721 1441 2161 2881 3601 4321
  ...
		

Crossrefs

Cf. A362778, A362779.

Cf. A038507 (1st column), A188914 (right diagonal).