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.

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, 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, 2, 5, 3, 5, 3, 3, 5, 3, 5, 7, 5, 5, 2, 7, 2, 3, 11, 3, 5, 3

Offset: 2

Author

Alain Rocchelli, Sep 04 2022

Keywords

• nonn

Comments

Conjecture: Such a prime(m) exists for every n, i.e., a(n) is never -1 for n>1.

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.

Examples

			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.
		

Links

• Alain Rocchelli, Table of n, a(n) for n = 2..10000

Crossrefs

Cf. A002110, A100380.

Programs

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

Python

from sympy import isprime, nextprime, prime
def a(n):
    pn, pm, pmsharp = prime(n), 2, 2
    while not isprime(pn + pmsharp): pm = nextprime(pm); pmsharp *= pm
    return pm
print([a(n) for n in range(2, 89)]) # Michael S. Branicky, Sep 04 2022

Formula

a(n) = prime(A100380(n)). - Michel Marcus, Sep 12 2022