A049782 a(n) = (0! + 1! + ... + (n-1)!) mod n.

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

Author

Clark Kimberling

Keywords

• nonn
• easy
• nice

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

• 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).

Crossrefs

Cf. A000142, A057245.

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

Programs

GAP

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

Haskell

a049782 :: Int -> Integer
a049782 n = (sum $ take n a000142_list) `mod` (fromIntegral n)
-- Reinhard Zumkeller, Nov 02 2011

Magma

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

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

Sage

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

Formula

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

Extensions

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