A360825 a(n) is the remainder after dividing n! by its least nondivisor.

1, 1, 2, 2, 4, 1, 6, 2, 5, 1, 10, 1, 12, 3, 8, 1, 16, 1, 18, 4, 11, 1, 22, 22, 6, 5, 14, 1, 28, 1, 30, 33, 20, 31, 18, 1, 36, 7, 20, 1, 40, 1, 42, 8, 23, 1, 46, 19, 11, 9, 26, 1, 52, 30, 27, 10, 29, 1, 58, 1, 60, 43, 53, 56, 33, 1, 66, 12, 35, 1, 70, 1, 72, 27, 23

Offset: 0

Author

Sebastian F. Orellana, Feb 22 2023

Keywords

• nonn

Comments

For every term besides a(3), the least nondivisor is the next prime after n.

Examples

			a(5) = 5! mod 7 = 120 mod 7 = 1.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..10000

Crossrefs

Cf. A000040, A000142, A006093, A040976, A151800, A213636, A275111, A360805.

Programs

Mathematica

a[n_] := Module[{f = n!, m = n + 1}, While[Divisible[f, m], m++]; Mod[f, m]]; Array[a, 100, 0] (* Amiram Eldar, Feb 22 2023 *)

PARI

a(n) = my(k=1, r); while(!(r=(n! % (n+k))), k++); r; \\ Michel Marcus, Feb 22 2023

Python

from functools import reduce
from sympy import nextprime
def A360825(n):
    if n == 3: return 2
    m = nextprime(n)
    return reduce(lambda i, j: i*j%m,range(2,n+1),1)%m # Chai Wah Wu, Feb 22 2023

Python

from functools import reduce
from sympy import nextprime
def A360825(n):
    if n == 3: return 2
    m = nextprime(n)
    return (m-1)*pow(reduce(lambda i,j:i*j%m,range(n+1,m),1),-1,m)%m # Chai Wah Wu, Feb 23 2023

Formula

a(n) = 1 <=> n in { A040976 } \ { 3 }.

a(n) = n <=> n in { A006093 }.

a(n) = n! mod A151800(n) for n > 3.

a(n) = A213636(n!) = A213636(A000142(n)).

a(A000040(n)) = A275111(n) for n >= 3.

a(n) > n <=> n in { A360805 }.