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Lexicographically earliest sequence of primes whose partial products lie between noncomposite numbers.
2

%I #20 Jul 08 2023 10:42:28

%S 2,2,3,5,3,13,5,7,41,13,83,109,347,337,127,67,379,499,739,4243,2311,

%T 1973,5827,7333,971,3449,3967,3407,12671,1423,859,20641,7237,769,9209,

%U 281,12919,16633,11383,30449,6733,40627,34591,1103,14303,5479,4603,17477,5113,51001,36299,57037,1153,34297,1237

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

%C 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 Michael S. Branicky, <a href="/A359948/b359948.txt">Table of n, a(n) for n = 1..160</a>

%e 2 - 1 = 1 and 2 + 1 = 3 are both noncomposites.

%e 2*2 - 1 = 3 and 2*2 + 1 = 5 are both primes.

%e 2*2*3 - 1 = 11 and 2*2*3 + 1 = 13 are both primes.

%e 2*2*3*5 - 1 = 59 and 2*2*3*5 + 1 = 61 are both primes.

%p R:= 2: s:= 2:

%p for i from 2 to 60 do

%p p:= 1:

%p do

%p p:= nextprime(p);

%p if isprime(p*s-1) and isprime(p*s+1) then R:= R,p; s:= p*s; break fi;

%p od od:

%p R;

%t 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 (Python)

%o from itertools import islice

%o from sympy import isprime, nextprime

%o def agen(): # generator of terms

%o s = 2; yield 2

%o while True:

%o p = 2

%o while True:

%o if isprime(s*p-1) and isprime(s*p+1):

%o yield p; s *= p; break

%o p = nextprime(p)

%o print(list(islice(agen(), 55))) # _Michael S. Branicky_, Jan 19 2023

%Y Cf. A036014, A359940.

%K nonn

%O 1,1

%A _Robert Israel_, Jan 19 2023