[math-fun] peculiar sequence
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
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] --
And lest you think it's a dreary down, down, down over large scales, check this out: 1000000000 77027750 0.0770278 2000000000 157560608 0.0787803 3000000000 220424252 0.0734748 4000000000 280562734 0.0701407 5000000000 321844298 0.0643689 6000000000 352149422 0.0586916 7000000000 457703806 0.0653863 8000000000 552286069 0.0690358 9000000000 615918561 0.0684354 10000000000 652538611 0.0652539 20000000000 1161806968 0.0580903 30000000000 1487066778 0.0495689 40000000000 2266954258 0.0566739 50000000000 2730734603 0.0546147 60000000000 3070175531 0.0511696 70000000000 3870566515 0.0552938 80000000000 4155919081 0.051949 90000000000 4383665137 0.0487074 100000000000 4945843957 0.0494584 200000000000 9600417374 0.0480021 On Fri, Sep 2, 2016 at 11:58 PM, Tomas Rokicki <rokicki@gmail.com> wrote:
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] --
-- -- http://cube20.org/ -- [ <http://golly.sf.net/>Golly link suppressed; ask me why] --
participants (2)
-
Bill Gosper -
Tomas Rokicki