Maybe it's finite, maybe it's not. Asymptotically of course the count of zeros in both decimal and hex in n! are n/4; surprising that. I could only get to 10^11 quickly: 1 10 9 0.9 2 100 64 0.64 3 1000 422 0.422 4 10000 3068 0.3068 5 100000 23617 0.23617 6 1000000 185894 0.185894 7 10000000 1234496 0.12345 8 100000000 10505970 0.10506 9 1000000000 77027750 0.0770278 10 10000000000 652538611 0.0652539 11 100000000000 4945843957 0.0494584 On Fri, Sep 2, 2016 at 10:22 PM, Bill Gosper <billgosper@gmail.com> wrote:
Despite my distaste for decimal digits, I noticed (while showing Mathematica to new user Miles) that 52! ends in twelve 0s both in decimal and hex. Equality happens for
Select[Range[0, 101], IntegerExponent[#!, 10] == IntegerExponent[#!, 16] &]
{0, 1, 2, 3, 4, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 20, 21, 22, 23, 28, 29, 32, 33, 35, 36, 37, 38, 39, 40, 41, 42, 43, 45, 46, 47, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 65, 70, 71, 76, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 90, 91, 95, 100, 101} with gradually declining frequency:
In[535]:= Count[Range@99999, n_ /; IntegerExponent[#, 10] == IntegerExponent[#, 16] &[n!]] // tim
During evaluation of In[535]:= 3610.464197,0
Out[535]= 23617
Is the asymptotic density finite? --rwg _______________________________________________ math-fun mailing list math-fun@mailman.xmission.com https://mailman.xmission.com/cgi-bin/mailman/listinfo/math-fun
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --