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%I #45 Mar 08 2021 22:14:29
%S 5,7,11,29,43,23,67,89,53,79,131,157,103,137,239,191,229,47,139,277,
%T 461,59,233,349,523,311,373,683,149,223,593,1259,83,739,821,1231,1559,
%U 173,431,947,1033,1291,1549,1721,283,659,941,1129,1223,1693,1787,2069,107,743,1061,1697,2333
%N Irregular triangle read by rows: row n gives the primes of the form m*prime(n)+1 where m is an even number <= prime(n) and prime(n) is the n-th prime, or 0 if no such prime exists for any n.
%C All safe primes are in this sequence.
%C 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.
%C 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
%H Metin Sariyar, <a href="/A330775/b330775.txt">Rows n = 1..220</a>
%F T(n, 1) = A035095(n) for n > 1. - _Michel Marcus_, Jan 02 2020
%e For n = 4, m = {4, 6}, prime(4) = 7, and 4*7+1 = 29, 6*7+1 = 43 are primes.
%e Rows of the triangle:
%e n=1 => {5}
%e n=2 => {7}
%e n=3 => {11}
%e n=4 => {29, 43}
%e n=5 => {23, 67, 89}
%e n=6 => {53, 79, 131, 157}
%e n=7 => {103, 137, 239}
%e n=8 => {191, 229}
%e n=9 => {47, 139, 277, 461}
%e ...
%t row[n_] := Select[2 * Range[Floor[(p = Prime[n])/2]] * p + 1, PrimeQ]; row /@ Range[16] //Flatten (* _Amiram Eldar_, Jan 02 2020 *)
%o (PARI) row(n) = select(x->isprime(x), vector(prime(n)\2, k, 2*k*prime(n)+1)); \\ _Michel Marcus_, Feb 05 2020
%Y Cf. A005384 (Sophie Germain primes), A005385 (safe primes), A035095.
%K nonn,tabf
%O 1,1
%A _Metin Sariyar_, Dec 30 2019
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