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%I #18 May 16 2018 11:33:22
%S 3,13,69,449,3248,25205,205022,1723507,14842906,130202808,1158787577,
%T 10433891463,94851898540,869200494599,8019346203785,74419210652835,
%U 694100859679691,6502464891216879,61154108320430275,577134533044522749,5463531774867094396
%N Largest number whose factorial is less than 10^(10^n).
%H Daniel Suteu, <a href="/A119906/b119906.txt">Table of n, a(n) for n = 0..500</a>
%e a(1) = 13 because 10^(10^1) = 10^10 = 10000000000 and 13! = 6227020800 is largest factorial less than this.
%p A119906Stirling := proc(n) local i; fsolve( (i+0.5)*log(i+1.0)=log(10.0)*10.0^n+i-1.0-0.5*log(2.0*Pi)-1./12.0/i,i) ; end : A119906 := proc(n::integer) local aestim::integer,resul::integer,faci1::integer,faci::integer,pow10::integer,i ; pow10 := 10^(10^n) ; aestim := floor(A119906Stirling(n))-1 ; faci1 := factorial(aestim) ; for i from aestim+1 to aestim+9 do faci := factorial(i) ; if faci1< pow10 and faci >= pow10 then RETURN(i-1) ; fi ; faci1 := faci ; end ; RETURN(0) ; end: for n from 0 to 30 do printf("%d,",A119906(n)) ; od ; # _R. J. Mathar_, Aug 04 2006
%t With[{s = Array[Factorial, 30000]}, Table[LengthWhile[s, # < 10^(10^n) &], {n, 0, 5}] ] (* _Michael De Vlieger_, May 14 2018 *)
%Y Cf. A000142, A061010, A152909.
%K nonn
%O 0,1
%A Julien Peter Benney (jpbenney(AT)ftml.net), Aug 01 2006
%E a(6) from _R. J. Mathar_, Aug 04 2006
%E a(7)-a(19) from _Jon E. Schoenfield_, Aug 09 2006
%E a(20) from _Daniel Suteu_, May 14 2018
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