OFFSET
3,2
COMMENTS
A theorem states that between (n+1)! + 2 and (n+1)! + (n+1) inclusive, there are n consecutive composite integers, namely 2, 3, 4, ..., n, n+1.
Records: 1, 3, 7, 27, 45, 95, 102, 127, 145, 146, 156, 225, 305, 420, 804, 844, 1173, 1671, 1725, 1827, 2570, 2930, 3318, 5142, 5946, 6837, 7007, 8208, 10221, ..., .
LINKS
Robert Israel, Table of n, a(n) for n = 3..607
FORMULA
EXAMPLE
For a(1), there are no positive primes which surround 1!. Therefore a(1) is undefined.
For a(2), there are two contiguous primes {2, 3} with 2 being 2!. The prime gap is 1. However, the two primes do not surround 2!, so a(2) is undefined.
For a(3), the following set of numbers, {5, 6, 7}, with 3! being in the middle. The prime gap is 2; therefore, a(3) = 1.
For a(4), the following set of numbers, {23, 24, 25, 26, 27, 28, 29} with 4! in between the two primes 23 & 29. The prime gap is 6, so a(4) = 3.
MAPLE
a:= n-> (f-> (nextprime(f-1)-prevprime(f+1))/2)(n!):
seq(a(n), n=3..70); # Alois P. Heinz, Jan 09 2021
MATHEMATICA
a[n_] := (NextPrime[n!, 1] - NextPrime[n!, -1])/2; Array[a, 70, 3]
PROG
(PARI) a(n) = (nextprime(n!+1) - precprime(n!-1))/2; \\ Michel Marcus, Jan 11 2021
(Python)
from sympy import factorial, nextprime, prevprime
def A340013(n):
f = factorial(n)
return (nextprime(f)-prevprime(f))//2 # Chai Wah Wu, Jan 23 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Robert G. Wilson v, Jan 09 2021
STATUS
approved