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%I #16 Jun 22 2021 01:02:36
%S 34,36,41,86,93,122,132,150,151,167,177,183,194,249,257,342,349,374,
%T 399,421,440,479,511,515,522,557,571,580,606,631,665,678,687,692,734,
%U 738,741,754,778,867,895,898,912,924,931,939,950,951,962,975,976,1012
%N Numbers whose factorial has '52' as its final two digits before the trailing zeros.
%H Charles R Greathouse IV, <a href="/A045561/b045561.txt">Table of n, a(n) for n = 1..10000</a>
%H <a href="/index/Fi#final">Index entries for sequences related to final digits of numbers</a>
%t Select[Range[5,1100],Take[Flatten[Most[Split[IntegerDigits[#!]]]],-2] == {5,2}&] (* _Harvey P. Dale_, Aug 08 2016 *)
%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)==52 \\ _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==52 && n>2 \\ _Charles R Greathouse IV_, Oct 22 2014
%o (PARI) v=List();m=Mod(6,25);for(n=4,1e6,m*=n/10^valuation(n,5);if(m==52,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_
%E More terms from _Robert G. Wilson v_, Dec 14 2000