A359948 Lexicographically earliest sequence of primes whose partial products lie between noncomposite numbers.

2, 2, 3, 5, 3, 13, 5, 7, 41, 13, 83, 109, 347, 337, 127, 67, 379, 499, 739, 4243, 2311, 1973, 5827, 7333, 971, 3449, 3967, 3407, 12671, 1423, 859, 20641, 7237, 769, 9209, 281, 12919, 16633, 11383, 30449, 6733, 40627, 34591, 1103, 14303, 5479, 4603, 17477, 5113, 51001, 36299, 57037, 1153, 34297, 1237

Offset: 1

Author

Robert Israel, Jan 19 2023

Keywords

• nonn

Comments

Are there any repeated terms other than a(1) = a(2) = 2, a(3) = a(5) = 3, a(4) = a(7) = 5 and a(6) = a(10) = 13?

Examples

			2 - 1 = 1 and 2 + 1 = 3 are both noncomposites.
2*2 - 1 = 3 and 2*2 + 1 = 5 are both primes.
2*2*3 - 1 = 11 and 2*2*3 + 1 = 13 are both primes.
2*2*3*5 - 1 = 59 and 2*2*3*5 + 1 = 61 are both primes.
		

Links

• Michael S. Branicky, Table of n, a(n) for n = 1..160

Crossrefs

Cf. A036014, A359940.

Programs

Maple

R:= 2: s:= 2:
for i from 2 to 60 do
  p:= 1:
  do
    p:= nextprime(p);
  if isprime(p*s-1) and isprime(p*s+1) then R:= R,p; s:= p*s; break fi;
od od:
R;

Mathematica

a[1] = 2; a[n_] := a[n] = Module[{r = Product[a[k], {k, 1, n - 1}], p = 2}, While[! PrimeQ[r*p - 1] || ! PrimeQ[r*p + 1], p = NextPrime[p]]; p]; Array[a, 55] (* Amiram Eldar, Jan 19 2023 *)

Python

from itertools import islice
from sympy import isprime, nextprime
def agen(): # generator of terms
    s = 2; yield 2
    while True:
        p = 2
        while True:
            if isprime(s*p-1) and isprime(s*p+1):
                yield p; s *= p; break
            p = nextprime(p)
print(list(islice(agen(), 55))) # Michael S. Branicky, Jan 19 2023