%I #18 Jun 22 2021 01:03:32
%S 61,96,130,139,159,161,166,178,187,192,211,218,236,255,264,324,334,
%T 341,361,388,406,432,446,458,469,483,489,494,507,537,549,555,575,576,
%U 587,624,637,654,657,684,691,697,746,756,788,793,797,824,838,840
%N Numbers whose factorial has '56' as its final two digits before the trailing zeros.
%H Harvey P. Dale, <a href="/A045562/b045562.txt">Table of n, a(n) for n = 1..1000</a>
%H <a href="/index/Fi#final">Index entries for sequences related to final digits of numbers</a>
%t lnzQ[n_]:=Module[{idn=IntegerDigits[n!],lz},lz=Length[Last[Split[idn]]];Take[idn,{-2-lz,-1-lz}]=={5,6}]; Select[Range[10,850],lnzQ] (* _Harvey P. Dale_, Mar 05 2013 *)
%o (PARI) last(n,k)=if(n*8\9-1<k, n=n!; return(n/10^valuation(n,10)%10^k)); my(m=Mod(1,5^k)); for(i=2,n,m*=i/10^valuation(i,5));lift(chinese(m, Mod(0,2^k))) \\ Gives the last k decimal digits of n!.
%o is(n)=last(n,2)==56 \\ _Charles R Greathouse IV_, Oct 22 2014
%o (PARI) is(n)=my(m=Mod(1,25)); for(i=2,n, m*=i/10^valuation(i,5)); m==56 \\ _Charles R Greathouse IV_, Oct 22 2014
%o (PARI) v=List();m=Mod(1,25);for(n=2,1e6,m*=n/10^valuation(n,5);if(m==56,listput(v,n);if(#v==100, return(Vec(v))))) \\ _Charles R Greathouse IV_, Oct 22 2014
%K nonn,base
%O 1,1
%A _Jeff Burch_