login

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).

A049782
a(n) = (0! + 1! + ... + (n-1)!) mod n.
10
0, 0, 1, 2, 4, 4, 6, 2, 1, 4, 1, 10, 10, 6, 4, 10, 13, 10, 9, 14, 13, 12, 21, 10, 14, 10, 10, 6, 17, 4, 2, 26, 1, 30, 34, 10, 5, 28, 10, 34, 4, 34, 16, 34, 19, 44, 18, 10, 48, 14, 13, 10, 13, 10, 34, 34, 28, 46, 28, 34, 22, 2, 55, 26, 49, 34, 65, 30, 67, 34, 68, 10, 55, 42, 64, 66, 34
OFFSET
1,4
COMMENTS
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
REFERENCES
R. K. Guy, Unsolved Problems in Number Theory, B44: is a(n) > 0 for n > 2?
LINKS
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.
FORMULA
a(n) = A003422(n) mod n = !n mod n. - G. C. Greubel, Dec 11 2019
MAPLE
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
MATHEMATICA
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 *)
PROG
(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
(Sage) [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
CROSSREFS
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
KEYWORD
nonn,easy,nice
EXTENSIONS
More terms from Erich Friedman, who observes that the first 500 terms are nonzero. Independently extended by Stephen A. Silver.
STATUS
approved