A067462 -id:A067462 - cp's OEIS frontend

A057245 Numbers m such that m | Sum_{k=1..m} k!.

1, 3, 9, 11, 33, 99

Offset: 1

Robert G. Wilson v, Sep 21 2000

n such that A067462(n) = 0. - T. D. Noe, May 13 2010
For any terms in this sequence, their LCM also belongs to this sequence. Term a(7), if it exists, is prime. - Max Alekseyev, Oct 14 2012
n > 1 is in this sequence iff A049782(n) = 1. - Max Alekseyev, Apr 17 2016
If it exists, a(7) > 12000000. - Bert Dobbelaere, Oct 28 2018
If it exists, a(7) > 1.6*10^8. - Giovanni Resta, Nov 09 2018

Richard K. Guy, Unsolved Problems in Number Theory, Springer, Third Ed., 2004, Section B44.

Cf. A007489, A049782, A064383.

Filtered([1..1500],m->Sum([1..m],k->Factorial(k)) mod m = 0); # Muniru A Asiru, Oct 28 2018

a = 0; b = 1; k = 2; While[k < 250001, c = k*b - (k - 1) a;
If[ Mod[c, k] == 1, Print[k]]; a = b; b = c; k++] (* Robert G. Wilson v, Jun 15 2013 *)

A100612 a(n) = (0! + 1! + ... + (p-1)!) mod p, where p = prime(n).

Original entry on oeis.org

0, 1, 4, 6, 1, 10, 13, 9, 21, 17, 2, 5, 4, 16, 18, 13, 28, 22, 65, 68, 55, 20, 27, 76, 80, 13, 50, 43, 65, 109, 56, 81, 93, 134, 82, 10, 131, 4, 30, 104, 29, 170, 104, 165, 9, 122, 130, 42, 225, 50, 69, 12, 128, 60, 147, 52, 16, 56, 7, 218, 154, 264, 198, 48, 299, 205, 251, 101

Offset: 1

N. J. A. Sloane, Dec 02 2004

nonn

The greedy inverse (indices of first occurrence of 1, 2, 3, ... in the sequence) is 2, 11, 91, 3, 12, 4, 59, -1, 8, 6, -1, 52, 7, 2550, -1, 14, 10, 15, 5461, 22, 9, 18, 205, 141, 4178, -1, 23, 17, 41, 39, -1, 5297, 937, -1, -1, -1, -1, 5248, 213, -1, 90, 48, 28, 4202, -1, 1718, 313, 64, 119, 27, ... where -1 means the number does not exist or is larger than 8000. - R. J. Mathar, Dec 19 2016
a(12397) = 31; a(54708) = 37. - Michel Marcus, May 11 2019
a(105527) = 35. - Michel Marcus, May 13 2019
a(16728884) = 26; a(62860131) = 35; sent by Milos Tatarevic. - Michel Marcus, May 18 2019

R. K. Guy, Unsolved Problems in Number Theory, B44: is a(n)>0 for n>2?

Michel Marcus, Table of n, a(n) for n = 1..2000
Vladica Andrejic and Milos Tatarevic, Searching for a counterexample to Kurepa's Conjecture, arXiv:1409.0800 [math.NT], 2014-2015.
Vladica Andrejic, Alin Bostan, Milos Tatarevic, Improved algorithms for left factorial residues, arXiv:1904.09196 [math.NT], 2019.
Luis H. Gallardo, Artin-Schreier, Erdős, and Kurepa’s conjecture, 2023.
Romeo Mestrovic, Variations of Kurepa's left factorial hypothesis, arXiv preprint arXiv:1312.7037 [math.NT], 2013-2014.
T. D. Noe, Plot of first 5000 terms (The red line gives prime(n). There are very few duplicate values in the sequence; the 5000 terms have 4476 values.)

See A049782 for more information. See also A003422, A236399.

Cf. A067462.

lf:=n->add(k!,k=0..n-1);
[seq(lf(ithprime(n)) mod ithprime(n),n=1..100)];
# 2nd program:
A100612 := proc(n)
    local p,f,a,k;
    f := 1 ;
    a := 0 ;
    p := ithprime(n) ;
    for k from 0 to p-1 do
        a := modp(a+f,p) ;
        f := modp(f*(k+1),p) ;
    end do:
    a ;
end proc:
seq(A100612(n),n=1..50) ; # R. J. Mathar, Dec 19 2016

Table[Mod[Total[Range[0,n-1]!],n],{n,Prime[Range[70]]}] (* Harvey P. Dale, May 06 2013 *)

a(n) = {my(p = prime(n), v = vector(p-1, k, Mod(k, p))); for (k=2, p-1, v[k] *= v[k-1];); lift(1+vecsum(v));} \\ Michel Marcus, May 05 2019

a(n) = A236399(n) mod prime(n).

a(n) = A067462(prime(n)) + 1, unless A067462(prime(n)) == - 1 (mod n). - Michel Marcus, May 05 2019

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A057245 Numbers m such that m | Sum_{k=1..m} k!.

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A100612 a(n) = (0! + 1! + ... + (p-1)!) mod p, where p = prime(n).

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