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A362777
Triangular array read by rows: T(n,k) = n!*k + 1, n >= 1, 1 <= k <= n.
2
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
1,1
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.
EXAMPLE
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. A038507 (1st column), A188914 (right diagonal).
Sequence in context: A231099 A062252 A153800 * A147791 A169647 A072467
KEYWORD
tabl,nonn
AUTHOR
Joe B. Stephen, May 03 2023
STATUS
approved