A362779 -id:A362779 - cp's OEIS frontend

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

Joe B. Stephen, May 03 2023

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.

			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
  ...

Cf. A362778, A362779.

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

A362778 Triangular array read by rows: T(n,k) is the least prime factor of n!*k + 1, n >= 1, 1 <= k <= n.

Original entry on oeis.org

2, 3, 5, 7, 13, 19, 5, 7, 73, 97, 11, 241, 19, 13, 601, 7, 11, 2161, 43, 13, 29, 71, 17, 15121, 20161, 11, 30241, 35281, 61, 11, 73, 161281, 449, 241921, 282241, 47, 19, 293, 1088641, 1451521, 23, 2177281, 13, 2903041, 17, 11, 13, 10886401, 233, 18144001, 17, 101, 29030401, 32659201, 43

Offset: 1

Joe B. Stephen, May 03 2023

The primes in each row are distinct because n!*k + 1 are coprime for 1 <= k <= n, and hence this array gives a simple proof that there are infinitely many prime numbers.

			Triangle T(n,k) begins:
  n\k  1    2    3    4    5    6 ...
  1    2
  2    3    5
  3    7   13   19
  4    5    7   73   97
  5   11  241   19   13  601
  6    7   11 2161   43   13   29
  ...

Cf. A362777, A362779.

Cf. A051301 (1st column).

T(n,k) = A020639(A362777(n,k)).

cp's OEIS Frontend

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

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