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

1,1

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?

Links

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

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A036014, A359940.

Sequence in context: A299995 A113167 A036014 * A289507 A076228 A317050

Adjacent sequences: A359945 A359946 A359947 * A359949 A359950 A359951

Keyword

nonn

Author

Robert Israel, Jan 19 2023

Status

approved