A008904 a(n) is the final nonzero digit of n!.
1, 1, 2, 6, 4, 2, 2, 4, 2, 8, 8, 8, 6, 8, 2, 8, 8, 6, 8, 2, 4, 4, 8, 4, 6, 4, 4, 8, 4, 6, 8, 8, 6, 8, 2, 2, 2, 4, 2, 8, 2, 2, 4, 2, 8, 6, 6, 2, 6, 4, 2, 2, 4, 2, 8, 4, 4, 8, 4, 6, 6, 6, 2, 6, 4, 6, 6, 2, 6, 4, 8, 8, 6, 8, 2, 4, 4, 8, 4, 6, 8, 8, 6, 8, 2, 2, 2, 4, 2, 8, 2, 2, 4, 2, 8, 6, 6, 2, 6
Offset: 0
Examples
6! = 720, so a(6) = 2.
References
- J.-P. Allouche and J. Shallit, Automatic Sequences, Cambridge Univ. Press, 2003, p. 202.
- Gardner, M. "Factorial Oddities." Ch. 4 in Mathematical Magic Show: More Puzzles, Games, Diversions, Illusions and Other Mathematical Sleight-of-Mind from Scientific American. New York: Vintage, pp. 50-65, 1978
- S. Kakutani, Ergodic theory of shift transformations, in Proc. 5th Berkeley Symp. Math. Stat. Prob., Univ. Calif. Press, vol. II, 1967, 405-414.
- Popular Computing (Calabasas, CA), Problem 120, Factorials, Vol. 4 (No. 36, Mar 1976), page PC36-3.
Links
- Robert G. Wilson v, Table of n, a(n) for n = 0..10000
- Washington Bomfim, An algorithm to find the last nonzero digit of n!
- Washington Bomfim, A property of the last non-zero digit of factorials
- K. S. Brown, The least significant nonzero digit of n!
- F. M. Dekking, Regularity and irregularity of sequences generated by automata, Sém. Théor. Nombres, Bordeaux, Exposé 9, 1979-1980, pages 9-01 to 9-10.
- Jean-Marc Deshouillers, A footnote to the least non zero digit of n! in base 12, Uniform Distribution Theory 7:1 (2012), pp. 71-73.
- Jean-Marc Deshouillers, Yet Another Footnote to the Least Non Zero Digit of n! in Base 12. Unif. Distrib. Theory 11 (2016), no. 2, 163-167.
- Gregory P. Dresden, Three transcendental numbers from the last non-zero digits of n^n, F_n and n!, Mathematics Magazine, pp. 96-105, vol. 81, 2008.
- Gregory P. Dresden, Two Irrational Numbers from the Last Nonzero Digits of n! and n^n, Mathematics Magazine, Vol. 74, No. 4 (2001), 316-320.
- Gregory P. Dresden, Research Papers.
- Fritz Jacob (fritzjacob(AT)gmail.com), A way to compute a(n)
- MathPages, Least Significant Non-Zero Digit of n!
- J. C. Martin, The structure of generalized Morse minimal sets on m symbols, Trans. Amer. Math. Soc. 232 (1977), 343-355.
- T. Sillke, What are the next entries for the following sequences? Puzzle U asks for the next number after 2642242888682886824484644846.
- Eric Weisstein's World of Mathematics, Factorial
- David W. Wilson, Minimal state machine for this sequence
- David W. Wilson, Another method for computing this sequence
- Index entries for sequences related to final digits of numbers
- Index entries for sequences related to factorial numbers
Crossrefs
Programs
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Haskell
a008904 n = a008904_list !! n a008904_list = 1 : 1 : f 2 1 where f n x = x' `mod` 10 : f (n+1) x' where x' = g (n * x) where g m | m `mod` 5 > 0 = m | otherwise = g (m `div` 10) -- Reinhard Zumkeller, Apr 08 2011 -
Mathematica
f[n_]:=Module[{m=n!},While[Mod[m,10]==0,m=m/10];Mod[m,10]] Table[f[i],{i,0,100}] f[n_] := Mod[6Times @@ (Rest[FoldList[{ 1 + #1[[1]], #2!2^(#1[[1]]#2)} &, {0, 0}, Reverse[IntegerDigits[n, 5]]]]), 10][[2]]; Join[{1, 1}, Table[f[n], {n, 2, 100}]] (* program contributed by Jacob A. Siehler, Greg Dresden, Feb 21 2006 *) zOF[n_Integer?Positive] := Module[{maxpow=0}, While[5^maxpow<=n,maxpow++]; Plus@@Table[Quotient[n,5^i], {i,maxpow-1}]]; Flatten[Table[ Take[ IntegerDigits[ n!], {-zOF[n]-1}],{n,100}]] (* Harvey P. Dale, Dec 16 2010 *) f[n_]:=Block[{id=IntegerDigits[n!, 10]}, While[id[[-1]]==0, id=Most@id]; id[[-1]]]; Table[f@n, {n, 0, 100}] (* Vincenzo Librandi, Sep 07 2017 *) -
PARI
a(n) = r=1; while(n>0, r *= Mod(4, 10)^((n\10)%2) * [1, 2, 6, 4, 2, 2, 4, 2, 8][max(n%10, 1)]; n\=5); lift(r) \\ Charles R Greathouse IV, Nov 05 2010; cleaned up by Max Alekseyev, Jan 28 2012
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Python
def a(n): if n <= 1: return 1 return 6*[1,1,2,6,4,4,4,8,4,6][n%10]*3**(n/5%4)*a(n/5)%10 # Maciej Ireneusz Wilczynski, Aug 23 2010 -
Python
from functools import reduce from sympy.ntheory.factor_ import digits def A008904(n): return reduce(lambda x,y:x*y%10,(((6,2,4,8,6,2,4,8,2,4,8,6,6,2,4,8,4,8,6,2)[(a<<2)|(i*a&3)] if i*a else (1,1,2,6,4)[a]) for i, a in enumerate(digits(n,5)[-1:0:-1])),6) if n>1 else 1 # Chai Wah Wu, Dec 07 2023
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Sage
def A008904(n): # algorithm from David Wilson, http://oeis.org/A008904/a008904b.txt if n == 0 or n == 1: return 1 dd = n.digits(base=5) x = sum(i*d for i,d in enumerate(dd)) y = sum(d for d in dd if d % 2 == 0)/2 z = 2**((x+y) % 4) if z == 1: z = 6 return z # D. S. McNeil, Dec 09 2010
Formula
The generating function for n>1 is as follows: for n = a_0 + 5*a_1 + 5^2*a_2 + ... + 5^N*a_N (the expansion of n in base-5), then the last nonzero digit of n!, for n>1, is 6*Product_{i=0..N} (a_i)! (2^(i a_i)) mod 10. - Greg Dresden, Feb 21 2006
a(n) = f(n,1,0) with f(n,x,e) = if n < 2 then A010879(x*A000079(e)) else f(n-1, A010879(x)*A132740(n), e+A007814(n)-A112765(n)). - Reinhard Zumkeller, Aug 16 2008
From Washington Bomfim, Jan 09 2011: (Start)
a(0) = 1, a(1) = 1, if n >= 2, with
n represented in base 5 as (a_h, ..., a_1, a_0)_5,
t = Sum_{i = h, h-1, ... , 0} (a_i even),
x = Sum_{i=h, h-1, ... , 1} (Sum_{k=h, h-1, ..., i}(a_i)),
z = (x + t/2) mod 4, and y = 2^z,
a(n) = 6*(y mod 2) + y*(1-(y mod 2)).
For n >= 5, and n mod 5 = 0,
i) a(n) = a(n+1) = a(n+3),
ii) a(n+2) = 2*a(n) mod 10, and
iii) a(n+4) = 4*a(n) mod 10.
For k not equal to 1, a(10^k) = a(2^k). See second Dresden link, and second Bomfim link.
(End)
Extensions
More terms from Greg Dresden, Feb 21 2006
Comments