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A210253 Number of distinct residues of all factorials mod 2^n.

%I A210253 #25 Nov 12 2020 12:05:17
%S A210253 1,2,3,4,5,6,8,8,9,10,11,13,14,16,16,16,17,18,19,20,22,24,24,25,26,27,
%T A210253 29,30,32,32,32,32,33,34,35,36,37,40,40,41,42,43,45,46,48,48,48,49,50,
%U A210253 51,52,54,56,56,57,58,59,61,62,64,64,64,64,64,65,66,67
%N A210253 Number of distinct residues of all factorials mod 2^n.
%C A210253 Theorem. For n>=1, a(n) = A007843(n) - A210255(n).
%H A210253 Alois P. Heinz, <a href="/A210253/b210253.txt">Table of n, a(n) for n = 0..1000</a>
%e A210253 Let n=2. We have modulo 4: 0!=1!==1, 2!==3!==2, for n>=4, n!==0. Thus the distinct residues are 0,1,2. Therefore, a(2) = 3.
%p A210253 a:= proc(n) local p, m, i, s;
%p A210253       p:= 2^n;
%p A210253       m:= 1;
%p A210253       s:= {};
%p A210253       for i to p while m<>0 do m:= m*i mod p; s:=s union {m} od;
%p A210253       nops(s)
%p A210253     end:
%p A210253 seq(a(n), n=0..100);  # _Alois P. Heinz_, Mar 20 2012
%t A210253 a[n_] := Module[{p = 2^n, m = 1, i, s = {}}, For[i = 1, i <= p && m != 0, i++, m = Mod[m i, p]; s = Union[s, {m}]]; Length[s]];
%t A210253 a /@ Range[0, 100] (* _Jean-François Alcover_, Nov 12 2020, after _Alois P. Heinz_ *)
%Y A210253 Cf. A000142, A007814, A210255.
%K A210253 nonn
%O A210253 0,2
%A A210253 _Vladimir Shevelev_, Mar 19 2012