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

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, 29, 30, 32, 32, 32, 32, 33, 34, 35, 36, 37, 40, 40, 41, 42, 43, 45, 46, 48, 48, 48, 49, 50, 51, 52, 54, 56, 56, 57, 58, 59, 61, 62, 64, 64, 64, 64, 64, 65, 66, 67

Offset

0,2

Comments

Theorem. For n>=1, a(n) = A007843(n) - A210255(n).

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1000

Example

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.

Maple

a:= proc(n) local p, m, i, s;
      p:= 2^n;
      m:= 1;
      s:= {};
      for i to p while m<>0 do m:= m*i mod p; s:=s union {m} od;
      nops(s)
    end:
seq(a(n), n=0..100);  # Alois P. Heinz, Mar 20 2012

Mathematica

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]];
a /@ Range[0, 100] (* Jean-François Alcover, Nov 12 2020, after Alois P. Heinz *)

Crossrefs

Cf. A000142, A007814, A210255.

Sequence in context: A379044 A256221 A359099 * A130916 A003965 A351988

Adjacent sequences: A210250 A210251 A210252 * A210254 A210255 A210256

Keyword

nonn

Author

Vladimir Shevelev, Mar 19 2012

Status

approved