OFFSET
1,1
COMMENTS
All safe primes are in this sequence.
Conjecture: For every prime p, there is at least one even m <= p such that m*p+1 is prime; this implies that no row is empty and there is no "0" in the sequence.
Conjecture: For every prime p, there is always a positive integer k <= p such that k*p+m is prime for any odd integer m, 0 < m < p. For example, for p = 11, k*11+m is prime for pairs {k,m}: {2,1}, {4,3}, {6,5}, {2,7}, {2,9}. - Metin Sariyar, Jan 26 2021
LINKS
Metin Sariyar, Rows n = 1..220
FORMULA
T(n, 1) = A035095(n) for n > 1. - Michel Marcus, Jan 02 2020
EXAMPLE
For n = 4, m = {4, 6}, prime(4) = 7, and 4*7+1 = 29, 6*7+1 = 43 are primes.
Rows of the triangle:
n=1 => {5}
n=2 => {7}
n=3 => {11}
n=4 => {29, 43}
n=5 => {23, 67, 89}
n=6 => {53, 79, 131, 157}
n=7 => {103, 137, 239}
n=8 => {191, 229}
n=9 => {47, 139, 277, 461}
...
MATHEMATICA
row[n_] := Select[2 * Range[Floor[(p = Prime[n])/2]] * p + 1, PrimeQ]; row /@ Range[16] //Flatten (* Amiram Eldar, Jan 02 2020 *)
PROG
(PARI) row(n) = select(x->isprime(x), vector(prime(n)\2, k, 2*k*prime(n)+1)); \\ Michel Marcus, Feb 05 2020
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Metin Sariyar, Dec 30 2019
STATUS
approved