A357680 a(n) is the number of primes that can be written as +-1! +- 2! +- 3! +- ... +- n!.

0, 1, 3, 4, 7, 11, 16, 29, 42, 72, 121, 191, 367, 693, 1215, 2221, 4116, 7577, 13900, 25634, 48322, 90046, 169016, 317819, 600982, 1138049, 2158939, 4103414, 7818761, 14923641, 28534404, 54624906, 104786140, 201233500, 386914300, 744876280, 1435592207

Offset: 1

Author

Zhining Yang, Oct 09 2022

Keywords

• nonn

Examples

			For n=4, a(4)=4 means there exist 4 solutions ([17, 19, 29, 31]) as follows:
  17 =  1! - 2! - 3! + 4!;
  19 = -1! + 2! - 3! + 4!;
  29 =  1! - 2! + 3! + 4!;
  31 = -1! + 2! + 3! + 4!.
		

Crossrefs

Cf. A000142, A059590, A089359.

Programs

Python

from sympy import isprime,factorial
def A357680(nmax):
    a=[0]
    t=[1]
    for n in range(2, nmax+1):
        k=factorial(n)
        s=[]
        for j in t:
            s.append(k-j)
            s.append(k+j)
        a.append(sum(1 for p in s if isprime(p)))
        t=s
    return(a)
print(A357680(21))

Python

from sympy import isprime
from math import factorial
from itertools import product
def a(n):
    f = [2*factorial(i) for i in range(1, n+1)]
    t = sum(f)//2
    return sum(1 for s in product([0, 1], repeat=n-1) if isprime(t-sum(f[i] for i in range(n-1) if s[i])))
print([a(n) for n in range(1, 20)]) # Michael S. Branicky, Oct 15 2022

Extensions

a(28)-a(30) from Michael S. Branicky, Oct 09 2022

a(31)-a(32) from Michael S. Branicky, Oct 10 2022

a(33)-a(34) from Michael S. Branicky, Oct 13 2022

a(35)-a(36) from Michael S. Branicky, Oct 26 2022

a(37) from Michael S. Branicky, Nov 13 2022