A136690 Final nonzero digit of n! in base 3.

1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2, 1, 1, 2, 2, 2, 1, 1, 1, 2, 1, 1, 2, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 2, 2, 1, 2, 2, 1, 1, 1, 2

Offset: 0

Author

Carl R. White, Jan 16 2008

Keywords

• base
• easy
• nonn

Examples

			6! = 720 decimal = 222200 ternary, so a(6) = 2.
		

Links

• Kevin Ryde, Table of n, a(n) for n = 0..10000
• Index entries for sequences that are fixed points of mappings

Crossrefs

Cf. A000142, A127110.

Other bases: A136691, A136692, A136693, A136694, A136695, A136696, A008904, A136697, A136698, A136699, A136700, A136701, A136702, A212307.

Programs

Mathematica

f[n_] := Mod[6 Times @@ (Rest[ FoldList[{1 + #1[[1]], #2! 2^(#1[[1]] #2)} &, {0, 0}, Reverse[ IntegerDigits[n, 3]]]]), 10][[2]]; # /. {0 -> 1} & /@ Mod[Table[f@n, {n, 0, 104}], 3] (* Robert G. Wilson v, Apr 17 2010 *)
fnzd[n_]:=Module[{sidn3=Split[IntegerDigits[n!,3]]},If[MemberQ[ Last[ sidn3],0], sidn3[[-2,1]], sidn3[[-1,1]]]]; Array[fnzd,110,0] (* Harvey P. Dale, May 03 2018 *)

PARI

a(n) = vecsum([bittest(220,b) |b<-digits(n,9)])%2 + 1; \\ Kevin Ryde, Dec 03 2022

Formula

From David Radcliffe, Sep 03 2021: (Start)

a(n) = (n! / A060828(n)) mod 3;

a(n) = 1 + (A189672(n) mod 2);

a(6*n) = a(6*n+1) = a(2*n);

a(6*n+2) = 3 - a(2*n);

a(6*n+3) = a(6*n+4) = 3 - a(2*n+1);

a(6*n+5) = a(2*n+1).

(End)

a(n) = A008904(A127110(n)). - Michel Marcus, Sep 04 2021

From Kevin Ryde, Dec 03 2022: (Start)

a(n) = 1 if n written in base 9 has an even number of digits {2,3,4,6,7}; and otherwise a(n) = 2.

Fixed point of the morphism 1 -> 1,1,2,2,2,1,2,2,1; 2 -> 2,2,1,1,1,2,1,1,2; starting from 1.

(End)

a(n) = A212307(n) mod 3. - Ridouane Oudra, Sep 25 2024

Extensions

More terms from Robert G. Wilson v, Apr 17 2010