A045547 Numbers whose factorial has '2' as its final nonzero digit.

2, 5, 6, 8, 14, 19, 34, 35, 36, 38, 40, 41, 43, 47, 50, 51, 53, 62, 67, 74, 84, 85, 86, 88, 90, 91, 93, 97, 109, 110, 111, 113, 115, 116, 118, 122, 129, 132, 145, 146, 148, 150, 151, 153, 162, 167, 174, 177, 180, 181, 183, 189, 194, 200, 201, 203, 212, 217

Offset: 1

Author

Jeff Burch

Keywords

• nonn
• base

Comments

From Robert Israel, Dec 16 2016: (Start)

If k is in the sequence, then:

if k == 0 (mod 5), k+1 is in the sequence;

if k == 1 (mod 5), k+1 is in A045548;

if k == 2 (mod 5), k+1 is in A045549;

if k == 3 (mod 5), k+1 is in A045550. (End)

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for sequences related to final digits of numbers

Crossrefs

Cf. A008904, A045548, A045549, A045550.

Programs

Maple

count:= 0:
r:= 1:
for n from 2 while count < 100 do
  r:= r*n;
  if r mod 10 = 0 then r:= r/10^padic:-ordp(r, 5) fi;
  if r mod 10 = 2 then count:= count+1; A[count]:= n fi;
od: seq(A[i], i=1..100); # Robert Israel, Dec 16 2016

Mathematica

f[ n_Integer, m_Integer ] := (c = 0; p = 1; While[ d = Floor[ n/5^p ]; d > 0, c = c + d; p++ ]; Mod[ n!/10^c, m ] ); Select[ Range[ 250 ], f[ #, 10 ] == 2 & ]
Join[{2},Select[Range[5,220],Most[Split[IntegerDigits[#!]]][[-1,1]] == 2&]] (* Harvey P. Dale, May 04 2016 *)
f[n_] := Mod[6 Times @@ (Rest[ FoldList[{1 + #1[[1]], #2! 2^(#1[[1]] #2)} &, {0, 0}, Reverse[ IntegerDigits[n, 5]]]]), 10][[2]] (* after Jacob A. Siehler & Greg Dresden in A008904 *); f[0] = f[1] = 1; Select[ Range[150], f[#] == 2 &] (* Robert G. Wilson v, Dec 28 2016 *)

PARI

lnz(n)=if(n<2, return(1)); my(m=Mod(1,5)); for(k=2,n, m*=k/10^valuation(k,5)); lift(chinese(Mod(0,2),m))
is(n)=lnz(n)==2 \\ Charles R Greathouse IV, Dec 16 2016

PARI

list(lim)=my(v=List(),m=Mod(1,5)); for(k=2,lim, m*=k/10^valuation(k,5); if(m==2, listput(v, k))); Vec(v) \\ Charles R Greathouse IV, Dec 16 2016

Python

from functools import reduce
from itertools import count, islice
from sympy.ntheory.factor_ import digits
def A045547_gen(startvalue=1): # generator of terms
    return filter(lambda n:2==reduce(lambda x,y:x*y%10,((1,1,2,6,4)[a]*((6,2,4,8)[i*a&3] if i*a else 1) for i, a in enumerate(digits(n,5)[-1:0:-1])))*6%10, count(max(startvalue,1)))
A045547_list = list(islice(A045547_gen(),30)) # Chai Wah Wu, Dec 07 2023