A393254 Smallest k such that both n! + k and n! + k + 2 are primes.

2, 2, 1, 5, 5, 17, 89, 59, 107, 71, 167, 17, 191, 347, 1667, 1457, 197, 281, 101, 137, 137, 149, 989, 4049, 4157, 1019, 311, 5519, 101, 2201, 7739, 6161, 6131, 3257, 9767, 13757, 9257, 2549, 4601, 7571, 13679, 6131, 6197, 7919, 599, 5279, 9929, 1061, 101, 4637, 1931, 14657

Offset

0,1

Comments

This sequence represents the distance from n! to the first twin prime pair following it.

Because n! acts as a strong Mertens filter (being divisible by all primes <= n), the local density of primes immediately following n! is significantly higher than in random intervals of similar magnitude.

Empirical observation by the author suggests that a(n) < n^3 for all n >= 2, which implies a highly compact bound for the first twin prime pair after a factorial. This sequence is the twin-prime analog to A037153.

Links

Robert Israel, Table of n, a(n) for n = 0..367

Formula

a(n) = min { k >= 0 | n! + k is prime and n! + k + 2 is prime }.

Example

For n = 3, 3! = 6. For k = 1, 2, 3, 4, the pairs (6+k, 6+k+2) are (7,9), (8,10), (9,11), and (10,12), none of which are twin primes. For k = 5, (11, 13) are twin primes, so a(3) = 5.
For n = 4, 4! = 24. For k = 1, 2, 3, 4, the numbers 24+k (25, 26, 27, 28) are all composite. For k = 5, (29, 31) are twin primes, so a(4) = 5.

Maple

f:= proc(n) local N, k;
   N:= n!;
   for k from 5 by 6 do
     if isprime(N+k) and isprime(N+k+2) then return k fi
   od
end proc:
f(0):= 2: f(1):= 2: f(2):= 1:
map(f, [$0..60]); # Robert Israel, Mar 08 2026

Mathematica

a[n_]:=Module[{k=0}, While[!PrimeQ[n!+k]||!PrimeQ[n!+k+2], k++]; k]; Array[a, 52, 0] (* Stefano Spezia, Mar 08 2026 *)

Prog

(Python)
from sympy import isprime, factorial
def A393254(n):
    fact_n = factorial(n)
    k = 0
    while True:
        if isprime(fact_n + k) and isprime(fact_n + k + 2):
            return k
        k += 1
print([A393254(n) for n in range(0, 16)])
(PARI) a(n) = my(k=0, f=n!); while (!(ispseudoprime(f+k) && ispseudoprime(f+k+2)), k++); k; \\ Michel Marcus, Mar 08 2026

Crossrefs

Cf. A037153, A033932.

Sequence in context: A133611 A010094 A019710 * A118806 A328646 A171670

Adjacent sequences: A393251 A393252 A393253 * A393255 A393256 A393257

Keyword

nonn

Author

Fan Yu, Mar 08 2026

Extensions

More terms from Michel Marcus, Mar 08 2026

Status

approved