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A362779 Triangular array read by rows: T(n,k) is the greatest prime factor of n!*k + 1, n >= 1, 1 <= k <= n.

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))

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)).

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

Original entry on oeis.org

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).

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User: Joe B. Stephen

A362779 Triangular array read by rows: T(n,k) is the greatest prime factor of n!*k + 1, n >= 1, 1 <= k <= n.

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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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Formula

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

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Author

Keywords

Comments

Examples

Crossrefs