A355329 Least increasing sequence of primes such that a(n) - 1 is a multiple of 6*n.

7, 13, 19, 73, 151, 181, 211, 241, 271, 421, 463, 577, 859, 1009, 1171, 1249, 1327, 1621, 2053, 2161, 2269, 2377, 3037, 3169, 3301, 3433, 3727, 4201, 5569, 5581, 5953, 6337, 6733, 7549, 7561, 7993, 9103, 9349, 9829, 10321, 10333, 10837, 11353, 11617, 12421, 12697, 12973, 13249, 14407, 15601

Offset: 1

Author

J. M. Bergot and Robert Israel, Jun 29 2022

Keywords

• nonn

Comments

a(n) is the least prime == 1 mod (6*n) and (for n >= 2) greater than a(n-1).

Examples

			a(5) = 151 because 151 is prime, 151-1 = 150 is divisible by 6*5, and 151 > a(4) = 73.
		

Links

• Robert Israel, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A070850.

Programs

Maple

A:= Vector(100):
A[1]:= 7:
for n from 2 to 100 do
  for k from floor((A[n-1]-1)/(6*n))+1  do
    p:= 6*n*k+1;
    if isprime(p) then A[n]:= p; break fi
od od:
convert(A,list);

Mathematica

a[n_] := a[n] = Module[{p = If[n == 1, 2, NextPrime[a[n - 1]]]}, While[!Divisible[p - 1, 6*n], p = NextPrime[p]]; p]; Array[a, 50] (* Amiram Eldar, Jun 29 2022 *)

Python

from itertools import count, islice
from sympy import nextprime
def A355329_gen(): # generator of terms
    p = 2
    for m in count(6,6):
        while q:=(p-1)%m:
            p = nextprime(p+m-q-1)
        yield p
        p = nextprime(p)
A355329_list = list(islice(A355329_gen(),30)) # Chai Wah Wu, Jun 30 2022