A305099 Least prime m such that either prime(n)# - m is prime or prime(n)# + m is prime, where p# denotes the product of all primes <= p.

3, 3, 7, 11, 13, 17, 19, 23, 37, 41, 67, 59, 47, 47, 67, 59, 61, 89, 89, 103, 79, 83, 89, 97, 103, 131, 113, 127, 223, 191, 163, 179, 389, 239, 151, 167, 173, 239, 199, 191, 199, 223, 233, 593, 293, 457, 227, 311, 373, 257, 307, 313, 283, 277, 271, 307, 307

Offset: 1

Author

David Nicolas Lopez, May 22 2018

Keywords

• nonn

Comments

Since it is known that the first 2000 terms of A005235 are primes, and the first 1000 terms of A055211 are primes, then the first 1000 terms of this sequence are also the least m > 1 such that prime(n)# - m is prime or prime(n)# + m is prime. - Amiram Eldar, Nov 02 2018

Examples

			For n = 6, the sixth primorial is 30030. The nearest prime such that p(6)# plus or minus prime equals its 30030's closest prime is equal to 17 because 30030+17=30047 which is prime or 30030 - 17 = 30013 which is also prime. Given that we only care about the smallest prime distance to the closest prime to the primorial, then we return 17.
For n = 7, the seventh primorial is 510510. The closest prime to the primorial is 510529 which is 510510 + 19; therefore 19 is in the sequence.
		

References

• Martin Gardner, The Last Recreations (1997), pp. 194-195.
• R. K. Guy, Unsolved Problems in Number Theory, Section A2
• Stephen P. Richards, A Number For Your Thoughts, 1982, p. 200.

Links

• S. W. Golomb, Evidence of Fortunes conjecture, Mathematics Magazine, Vol. 54, No. 4 (Sep., 1981), pp. 209-210.
• The Prime Glossary, Fortunate and lesser fortunate numbers

Crossrefs

Cf. A005235, A055211.

Programs

Mathematica

primorial[n_] := Product[Prime[i], {i, 1, n}]; a[n_] := Module[{k = 2, pr = primorial[n]}, While[! PrimeQ[pr - k] && ! PrimeQ[pr + k], k = NextPrime[k]]; k]; Array[a, 57] (* Amiram Eldar, Oct 31 2018 *)

PARI

a(n) = { my(pr = prod(k=1, n, prime(k)), m=2); while (!isprime(pr-m) && !isprime(pr+m), m = nextprime(m+1)); m;} \\ Michel Marcus, Nov 02 2018

Sage

# returns quasi-fortunate-numbers up to n
def generateQFN(n):
    quasi_fortunate_numbers = []
    primorialArray = []
    prime = Primes()
    num_length = n+1
    primorial = 1
    for i in range(num_length):
        primorial *= prime[i]
        primorialArray.append(primorial)
    for primorials in primorialArray:
        num = 0
        while num < num_length:
            if is_prime(primorials+prime[num]):
                quasi_fortunate_numbers.append(prime[num])
                break
            elif is_prime(primorials-prime[num]):
                quasi_fortunate_numbers.append(prime[num])
                break
            num += 1
    return quasi_fortunate_numbers
generateQFN(7)

Formula

a(n) = min(A005235(n), A055211(n)), for n > 1.

Extensions

More terms from Amiram Eldar, Oct 31 2018