A086330 -id:A086330 - cp's OEIS frontend

A371035 a(n) = A086330(prime(n)).

0, 2, 7, 18, 43, 73, 113, 159, 203, 334, 496, 706, 863, 874, 1097, 1124, 1560, 2033, 2073, 2409, 2462, 3336, 3345, 3634, 3958, 4657, 5198, 5284, 5186, 6096, 7801, 8594, 9270, 9167, 10659, 10578, 12375, 12227, 13221, 13769, 15958, 16458, 18820, 17919, 18722

Offset: 1

Alexandre Herrera, Apr 10 2024

The sequence sometimes decreases, as for example at a(29) = 5186 < 5284 = a(28).

			For n = 3, a(n) = A086330(prime(3)) = A086330(5) = (2! mod 5) + (3! mod 5) + (4! mod 5) = 2 + 1 + 4 = 7.

Cf. A086330, A000040.

a(n) = my(p=prime(n)); sum(m=2, p, m! % p); \\ Michel Marcus, Apr 11 2024

from sympy import isprime
l = []
for i in range(2,185):
    if isprime(i):
        sum = 0
        reminder = 1
        for j in range(2, i):
            reminder = (reminder * j) % i
            sum += reminder
        l.append(sum)
print(l)

from sympy import prime
def A371035(n):
    a, c, p = 0, 1, prime(n)
    for m in range(2,p):
        c = c*m%p
        a += c
    return a # Chai Wah Wu, Apr 16 2024

A226570 a(n) = Sum_{k=1..n} (k+1)! mod n.

Original entry on oeis.org

0, 0, 2, 0, 2, 2, 4, 0, 8, 2, 10, 8, 8, 4, 2, 8, 11, 8, 7, 12, 11, 10, 19, 8, 12, 8, 8, 4, 15, 2, 0, 24, 32, 28, 32, 8, 3, 26, 8, 32, 2, 32, 14, 32, 17, 42, 16, 8, 46, 12, 11, 8, 11, 8, 32, 32, 26, 44, 26, 32, 20, 0, 53, 24, 47, 32, 63, 28, 65, 32, 66, 8, 53, 40, 62, 64, 32, 8, 18, 72, 62, 2, 25, 32, 62, 14, 44, 32, 74, 62, 60, 88, 62, 16, 7, 56, 78, 46, 98

Offset: 1

M. F. Hasler, Jun 11 2013

nonn

Motivated by sequence A100083, numbers such that a(n) = 0.

Note that this is a different sequence from A086330: in this one, the factorials are added up and then the remainder of the total divided by n is taken, whereas in A086330 each factorial is computed modulo n prior to being added up. - Alonso del Arte, Jun 11 2013

			a(3) = 2 because 2!, 3! and 4! are 2, 6 and 24 respectively, which add up to 32, and modulo 3 that is 2.
a(4) = 0 because 2!, 3!, 4! and 5! add up to 152, and modulo 4 that is 0 (note that this is different from A086330(4) = 4).

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Cf. A002034, A086330, A100083, A007489, A003422.

Table[Mod[Sum[(k + 1)!, {k, n}], n], {n, 75}] (* Alonso del Arte, Jun 11 2013 *)

PARI
```
a(n)=lift(sum(m=2,n-1,Mod(m!,n)))
```

a(n)=my(t=Mod(1,n)); lift(sum(m=2,n+1,t*=m)) \\ Charles R Greathouse IV, Jun 11 2013

def A226570(n):
    a, c = 0, 1
    for m in range(2,n):
        c = c*m%n
        if c==0:
            break
        a = (a+c)%n
    return a # Chai Wah Wu, Apr 16 2024

a(n) = A086330(n) mod n. - Chai Wah Wu, Apr 16 2024

a(n) = Sum_{k=1..A002034(n)-2} (k+1)! mod n. - David A. Corneth, Apr 16 2024

cp's OEIS Frontend

A371035 a(n) = A086330(prime(n)).

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A226570 a(n) = Sum_{k=1..n} (k+1)! mod n.

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