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

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

1,3

Comments

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

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

Links

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

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A029906 A094907 A343401 * A158380 A051734 A157898

Adjacent sequences: A226567 A226568 A226569 * A226571 A226572 A226573

Keyword

nonn

Author

M. F. Hasler, Jun 11 2013

Status

approved