A045550 Numbers whose factorial has '8' as its final nonzero digit.

9, 10, 11, 13, 15, 16, 18, 22, 27, 30, 31, 33, 39, 44, 54, 57, 70, 71, 73, 77, 80, 81, 83, 89, 94, 102, 105, 106, 108, 114, 119, 125, 126, 128, 137, 142, 149, 154, 157, 170, 171, 173, 184, 185, 186, 188, 190, 191, 193, 197, 204, 207, 220, 221, 223

Offset: 1

Author

Jeff Burch

Keywords

• nonn
• base

Comments

From Robert Israel, Dec 16 2016: (Start)

If n is in the sequence, then:

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

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

if n == 2 (mod 5), n+1 is in A045548;

if n == 3 (mod 5), n+1 is in A045547. (End)

Links

• Robert Israel, Table of n, a(n) for n = 1..10000 (n=1..1000 from Harvey P. Dale)
• Index entries for sequences related to final digits of numbers

Crossrefs

Cf. A008904, A045547, A045548, A045549.

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 = 8 then count:= count+1; A[count]:= n fi;
od:
seq(A[i], i=1..100); # Robert Israel, Dec 16 2016

Mathematica

Select[Range[5,250],Last[Flatten[Most[Split[IntegerDigits[#!]]]]]==8&] (* Harvey P. Dale, Jun 11 2014 *)
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[#] == 8 &] (* Robert G. Wilson v, Dec 28 2016 *)

Python

from itertools import count, islice
from functools import reduce
from sympy.ntheory.factor_ import digits
def A045550_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:8==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), count(max(startvalue,1)))
A045550_list = list(islice(A045550_gen(),30)) # Chai Wah Wu, Dec 07 2023