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%I #9 May 13 2013 00:30:39
%S 9,45,3,225,18,15,-1,1125,1,99,495,33,2475,198,165,-1,12375,11,999,
%T 4995,333,24975,1998,1665,-1,124875,111,9999,49995,3333,249975,19998,
%U 16665,-1,1249875,1111,99999,49995,33333,2499975,199998,166665,-1,12499875,11111,999999,4999995,333333,24999975,1999998,1666665,-1,124999875,111111
%N Murai Chuzen numbers.
%C "Murai Chuzen divides 9 by 1, 2, 3, 4, 5, 6, 7, 8, 9, getting the figures 9, 45, 3, 225, 18, 15, x (not divisible), 1125, 1, -- without reference to the decimal points. Similarly he divides 99 by 1, 2, 3, 4, 5, 6, 7, 8, 9, getting the figures 99, 495, 33, 2475, 198, 165, x, 12375, 11. Next he divides 999 by 1, 2, 3, 4, 5, 6, 7, 8, 9, getting the figures 999, 4995, 333, 24975, 1998, 1665, x, 124875, 111." (Smith and Mikami, expanded and corrected)
%C Smith and Mikami put "x" whenever a decimal does not terminate. In the data, I put -1 instead of "x".
%C Murai Chuzen concludes that if 1 is divided by 9, 45, 3, 225, 18, 15, 1125, and 1, the results will have one-digit repetends; if 1 is divided by 99, 495, 33, 2475, 198, 165, 12375, and 11, the results will have two-digit repetends; if 1 is divided by 999, 4995, 333, 24975, 1998, 1665, 124875, and 111, the results will have three-digit repetends; etc.
%D Murai Chuzen, Sampo Doshi-mon (Arithmetic for the Young), 1781.
%H David Eugene Smith and Yoshio Mikami, <a href="http://books.google.com/books?id=J1YNAAAAYAAJ&pg=PA176&lpg=PA176&dq=%22his+theory+is+brief%22&source=bl&ots=SXl7pxSOKl&sig=we-Nb2Ih5F6QxT2LTPAWQxfhqaQ&hl=en&sa=X&ei=8wmNUaqsO8jr0QGzxIDwBg&ved=0CBoQ6AEwAQ#v=onepage&q=%22his%20theory%20is%20brief%22&f=false">A history of Japanese mathematics</a>, Open Court, 1914, reprinted by Dover, 2004, p. 176.
%e 9/1 = 9, so a(1) = 9; 9/2 = 4.5, so a(2) = 45; 9/7 does not terminate, so a(7) = -1; 9/8 = 1.125, so a(8) = 1125; 9/9 = 1, so a(9) = 1.
%e 99/1 = 99, so a(10) = 99; 99/2 = 49.5, so a(11) = 495.
%Y Cf. A001913, A007732, A066799, A096688, A121090, A121341, A181431.
%K base,sign
%O 1,1
%A _Jonathan Sondow_, May 10 2013