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A356741 a(n) is the least prime(m) such that prime(n) + prime(m)# is prime, where prime(m)# denotes the product of the first m primes, or -1 if no such prime(m) exists.

%I A356741 #59 Oct 18 2022 11:21:08
%S A356741 2,2,3,2,3,2,7,3,2,3,3,2,5,3,3,2,3,3,2,3,5,3,11,3,2,3,2,5,11,5,3,2,7,
%T A356741 2,3,3,5,3,3,2,5,2,3,2,5,5,3,2,7,3,2,5,3,3,3,2,3,3,2,5,7,3,2,7,5,3,5,
%U A356741 2,5,3,5,3,3,5,3,5,7,5,5,2,7,2,3,11,3,5,3
%N A356741 a(n) is the least prime(m) such that prime(n) + prime(m)# is prime, where prime(m)# denotes the product of the first m primes, or -1 if no such prime(m) exists.
%C A356741 Conjecture: Such a prime(m) exists for every n, i.e., a(n) is never -1 for n>1.
%C A356741 Conjecture: Limit_{N->oo} (Sum_{n=2..N} a(n)) / (Sum_{n=2..N} log(prime(n))) = C with C constant between 0.5 and 1 inclusive.
%H A356741 Alain Rocchelli, <a href="/A356741/b356741.txt">Table of n, a(n) for n = 2..10000</a>
%F A356741 a(n) = prime(A100380(n)). - _Michel Marcus_, Sep 12 2022
%e A356741 For n=4, prime(4)=7, and m=1 gives prime(m)=2 and prime(n) + prime(m)# = 7 + 2 = 9 (nonprime), but m=2 gives prime(m)=3 and prime(n) + prime(m)# = 7 + 2*3 = 13 (prime), so a(4) = prime(2) = 3.
%o A356741 (Python)
%o A356741 from sympy import isprime, nextprime, prime
%o A356741 def a(n):
%o A356741     pn, pm, pmsharp = prime(n), 2, 2
%o A356741     while not isprime(pn + pmsharp): pm = nextprime(pm); pmsharp *= pm
%o A356741     return pm
%o A356741 print([a(n) for n in range(2, 89)]) # _Michael S. Branicky_, Sep 04 2022
%o A356741 (PARI) a(n) = my(p=2, pr=2, pn=prime(n)); while (!isprime(pn+pr), p=nextprime(p+1); pr *= p); p; \\ _Michel Marcus_, Sep 05 2022
%Y A356741 Cf. A002110, A100380.
%K A356741 nonn
%O A356741 2,1
%A A356741 _Alain Rocchelli_, Sep 04 2022