A330775 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.

5, 7, 11, 29, 43, 23, 67, 89, 53, 79, 131, 157, 103, 137, 239, 191, 229, 47, 139, 277, 461, 59, 233, 349, 523, 311, 373, 683, 149, 223, 593, 1259, 83, 739, 821, 1231, 1559, 173, 431, 947, 1033, 1291, 1549, 1721, 283, 659, 941, 1129, 1223, 1693, 1787, 2069, 107, 743, 1061, 1697, 2333

Offset: 1

Author

Metin Sariyar, Dec 30 2019

Keywords

• nonn
• tabf

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

Examples

			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}
...
		

Links

• Metin Sariyar, Rows n = 1..220

Crossrefs

Cf. A005384 (Sophie Germain primes), A005385 (safe primes), A035095.

Programs

Mathematica

row[n_] := Select[2 * Range[Floor[(p = Prime[n])/2]] * p + 1, PrimeQ]; row /@ Range[16] //Flatten (* Amiram Eldar, Jan 02 2020 *)

PARI

row(n) = select(x->isprime(x), vector(prime(n)\2, k, 2*k*prime(n)+1)); \\ Michel Marcus, Feb 05 2020

Formula

T(n, 1) = A035095(n) for n > 1. - Michel Marcus, Jan 02 2020