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

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

Author

N. J. A. Sloane, Dec 02 2004

Keywords

• nonn

Comments

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

References

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

Links

• 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.)

Crossrefs

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

Cf. A067462.

Programs

Maple

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

Mathematica

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

PARI

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

Formula

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