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A100612 a(n) = (0! + 1! + ... + (p-1)!) mod p, where p = prime(n). 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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, 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.

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.)

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

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 *)

PROG

(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

CROSSREFS

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

Cf. A067462.

Sequence in context: A119439 A290823 A090642 * A322778 A079160 A230256

Adjacent sequences:  A100609 A100610 A100611 * A100613 A100614 A100615

KEYWORD

nonn,changed

AUTHOR

N. J. A. Sloane, Dec 02 2004

STATUS

approved

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Last modified May 20 12:33 EDT 2019. Contains 323422 sequences. (Running on oeis4.)