A362777 -id:A362777 - cp's OEIS frontend

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

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)).

A362779 Triangular array read by rows: T(n,k) is the greatest 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, 37, 601, 103, 131, 2161, 67, 277, 149, 71, 593, 15121, 20161, 79, 30241, 35281, 661, 7331, 1657, 161281, 449, 241921, 282241, 6863, 269, 2477, 1088641, 1451521, 78887, 2177281, 5281, 2903041, 192113, 329891, 29383, 10886401, 62297, 18144001, 2243, 251501, 29030401, 32659201, 843907

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   37  601
  6   103  131 2161   67  277  149
  ...

Cf. A362777, A362778.

Cf. A002583 (1st column).

T(n,k) = A006530(A362777(n,k))

cp's OEIS Frontend

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

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