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%I #30 Sep 14 2019 06:36:12
%S 2,3,4,5,6,10,12,12,17,19,21,26,29,26,31,35,37,41,42,39,44,49,55,59,
%T 59,65,71,75,63,73,80,82,90,90,104,86,103,104,107,111,113,114,120,125,
%U 120,115,139,149,132,141,147,150,147,164,166,172,172,170,172,180
%N Number of distinct residues of all factorials mod prime(n).
%C Conjecture: a(n)/p_n > 1/2.
%C The standard (folklore?) conjecture is that a(n)/prime(n) = 1 - 1/e = 0.63212.... - _Charles R Greathouse IV_, May 11 2015
%H Amiram Eldar, <a href="/A210184/b210184.txt">Table of n, a(n) for n = 1..10000</a> (terms 1...1000 from Alois P. Heinz)
%H Yong-Gao Chen and Li-Xia Dai, <a href="http://emis.impa.br/EMIS/journals/INTEGERS/papers/g21/g21.Abstract.html">Congruences with factorials modulo p</a>, INTEGERS 6 (2006), #A21.
%e Let n=4, p_4=7. We have modulo 7: 1!==1, 2!==2, 3!==6, 4!==3, 5!==1, 6!==6 and for m>=7, m!==0, such that we have 5 distinct residues 0,1,2,3,6. Therefore a(4) = 5.
%p a:= proc(n) local p, m, i, s;
%p p:= ithprime(n);
%p m:= 1;
%p s:= {};
%p for i to p do m:= m*i mod p; s:=s union {m} od;
%p nops(s)
%p end:
%p seq(a(n), n=1..100); # _Alois P. Heinz_, Mar 19 2012
%t Table[Length[Union[Mod[Range[Prime[n]]!, Prime[n]]]], {n, 100}] (* _T. D. Noe_, Mar 18 2012 *)
%o (PARI) apply(p->#Set(vector(p,n,n!)%p), primes(100)) \\ _Charles R Greathouse IV_, May 11 2015
%o (PARI) a(n,p=prime(n))=my(t=1); #Set(vector(p,n,t=(t*n)%p)) \\ _Charles R Greathouse IV_, May 11 2015
%Y Cf. A000040, A000142.
%K nonn
%O 1,1
%A _Vladimir Shevelev_, Mar 18 2012
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