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

1,1

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

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

Sequence in context: A057247 A157437 A213677 * A340750 A375096 A031134

Adjacent sequences: A330772 A330773 A330774 * A330776 A330777 A330778

Keyword

nonn,tabf

Author

Metin Sariyar, Dec 30 2019

Status

approved