A359940 -id:A359940 - cp's OEIS frontend

A359939 Lexicographically earliest strictly increasing sequence of primes whose partial products lie between noncomposite numbers.

2, 3, 5, 19, 41, 67, 113, 653, 883, 1439, 3823, 10631, 12841, 14251, 23357, 27103, 30491, 64679, 78823, 110977, 115127, 118747, 159431, 215587, 301039, 342257, 343639, 428401, 473383, 493583, 566723, 621133, 638371, 639157, 680539, 904049, 993037, 1146133, 1252507

Offset: 1

Amiram Eldar, Jan 19 2023

nonn

			2 - 1 = 1 and 2 + 1 = 3 are both noncomposite numbers.
2*3 - 1 = 5 and 2*3 + 1 = 7 are both noncomposite numbers.
2*3*5 - 1 = 29 and 2*3*5 + 1 = 31 are both noncomposite numbers.

Amiram Eldar, Table of n, a(n) for n = 1..150

Cf. A008578, A014574, A039726, A087898, A167777, A359940.

a[1] = 2; a[n_] := a[n] = Module[{r = Product[a[k],{k, 1, n-1}], p = NextPrime[a[n-1]]}, While[!PrimeQ[r*p-1] || !PrimeQ[r*p+1], p = NextPrime[p]]; p]; Array[a, 50]

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

Original entry on oeis.org

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

Robert Israel, Jan 19 2023

nonn

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?

			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.

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

Cf. A036014, A359940.

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;

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 *)

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

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A359939 Lexicographically earliest strictly increasing sequence of primes whose partial products lie between noncomposite numbers.

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