1,4

Kurepa's conjecture is that gcd(!n,n!) = 2, n > 1. It is easy to prove that this is equivalent to showing that gcd(p,!p) = 1 for all odd primes p. In Guy, 2nd edition, it is stated that Mijajlovic has tested up to p = 10^6. Subsequently Gallot tested up to 2^26. I have continued up to just above p = 2^27, in fact to p < 144000000. There were no examples found where gcd(p,!p) > 1. - Paul Jobling, Dec 02 2004

According to Kellner, the conjecture has been proved by Barsky and Benzaghou. - T. D. Noe, Dec 02 2004

Barsky and Benzaghou withdrew their proof in 2011. I've extended the search up to 10^9; no counterexample was found. - Milos Tatarevic, Feb 01 2013

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

T. D. Noe, Table of n, a(n) for n = 1..10000

Romeo Mestrovic, The Kurepa-Vandermonde matrices arising from Kurepa's left factorial hypothesis, Filomat 29:10 (2015), 2207-2215. DOI:10.2298/FIL1510207M

Vladica Andrejic, Milos Tatarevic, Searching for a counterexample of Kurepa's Conjecture, arXiv:1409.0800 [math.NT], 2014.

D. Barsky and B. Benzaghou, Nombres de Bell et somme de factorielles, Journal de Théorie des Nombres de Bordeaux, 16:1, No. 17, 2004.

D. Barsky and B. Benzaghou, Erratum à l'article "Nombres de Bell et somme de factorielles", Journal de Théorie des Nombres de Bordeaux, 23:2 (2011), p. 527.

Y. Gallot, More information.

Bernd C. Kellner, Some remarks on Kurepa's left factorial, arXiv:math/0410477 [math.NT], 2004.

Romeo Mestrovic, Variations of Kurepa's left factorial hypothesis, arXiv preprint arXiv:1312.7037 [math.NT], 2013-2014.

Stephen A. Silver, C program to generate this sequence.

M. Tatarevic, Searching for a counterexample to the Kurepa's left factorial hypothesis (p < 10^9).

a(n) = A003422(n) mod n = !n mod n. - G. C. Greubel, Dec 11 2019

a:= proc(n) local c, i, t; c, t:=1, 1;

for i to n-1 do t:= (t*i) mod n; c:= c+t od; c mod n

end:

seq(a(n), n=1..100); # Alois P. Heinz, Feb 16 2013

Table[Mod[Sum[ i!, {i, 0, n-1}], n], {n, 80}]

nn=80; With[{fcts=Accumulate[Range[0, nn]!]}, Flatten[Table[Mod[Take[fcts, {n}], n], {n, nn}]]] (* Harvey P. Dale, Sep 22 2011 *)

(Haskell)

a049782 :: Int -> Integer

a049782 n = (sum $ take n a000142_list) `mod` (fromIntegral n)

-- Reinhard Zumkeller, Nov 02 2011

(PARI) a(n)=my(s=1, f=1); for(k=1, n, f=f*k%n; s+=f); s%n \\ Charles R Greathouse IV, Feb 07 2017

(Magma) [&+[Factorial(k-1): k in [1..n]] mod (n): n in [1..80]]; // Vincenzo Librandi, May 31 2019

(SageMath) [mod(sum(factorial(k) for k in (0..n-1)), n) for n in (1..80)] # G. C. Greubel, Dec 11 2019

(GAP) List([1..80], n-> Sum([0..n-1], k-> Factorial(k)) mod n ); # G. C. Greubel, Dec 11 2019

Cf. A000142, A057245.

Note that in the context of this sequence, !n is the left factorial A003422 not the subfactorial A000166.

Sequence in context: A306340 A205969 A326771 * A372785 A091666 A084290

Adjacent sequences: A049779 A049780 A049781 * A049783 A049784 A049785

nonn,easy,nice

More terms from Erich Friedman, who observes that the first 500 terms are nonzero. Independently extended by Stephen A. Silver.

approved