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

%I A359948 #20 Jul 08 2023 10:42:28
%S A359948 2,2,3,5,3,13,5,7,41,13,83,109,347,337,127,67,379,499,739,4243,2311,
%T A359948 1973,5827,7333,971,3449,3967,3407,12671,1423,859,20641,7237,769,9209,
%U A359948 281,12919,16633,11383,30449,6733,40627,34591,1103,14303,5479,4603,17477,5113,51001,36299,57037,1153,34297,1237
%N A359948 Lexicographically earliest sequence of primes whose partial products lie between noncomposite numbers.
%C A359948 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?
%H A359948 Michael S. Branicky, <a href="/A359948/b359948.txt">Table of n, a(n) for n = 1..160</a>
%e A359948 2 - 1 = 1 and 2 + 1 = 3 are both noncomposites.
%e A359948 2*2 - 1 = 3 and 2*2 + 1 = 5 are both primes.
%e A359948 2*2*3 - 1 = 11 and 2*2*3 + 1 = 13 are both primes.
%e A359948 2*2*3*5 - 1 = 59 and 2*2*3*5 + 1 = 61 are both primes.
%p A359948 R:= 2: s:= 2:
%p A359948 for i from 2 to 60 do
%p A359948   p:= 1:
%p A359948   do
%p A359948     p:= nextprime(p);
%p A359948   if isprime(p*s-1) and isprime(p*s+1) then R:= R,p; s:= p*s; break fi;
%p A359948 od od:
%p A359948 R;
%t A359948 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 *)
%o A359948 (Python)
%o A359948 from itertools import islice
%o A359948 from sympy import isprime, nextprime
%o A359948 def agen(): # generator of terms
%o A359948     s = 2; yield 2
%o A359948     while True:
%o A359948         p = 2
%o A359948         while True:
%o A359948             if isprime(s*p-1) and isprime(s*p+1):
%o A359948                 yield p; s *= p; break
%o A359948             p = nextprime(p)
%o A359948 print(list(islice(agen(), 55))) # _Michael S. Branicky_, Jan 19 2023
%Y A359948 Cf. A036014, A359940.
%K A359948 nonn
%O A359948 1,1
%A A359948 _Robert Israel_, Jan 19 2023