

A225488


Murai Chuzen numbers.


0



9, 45, 3, 225, 18, 15, 1, 1125, 1, 99, 495, 33, 2475, 198, 165, 1, 12375, 11, 999, 4995, 333, 24975, 1998, 1665, 1, 124875, 111, 9999, 49995, 3333, 249975, 19998, 16665, 1, 1249875, 1111, 99999, 49995, 33333, 2499975, 199998, 166665, 1, 12499875, 11111, 999999, 4999995, 333333, 24999975, 1999998, 1666665, 1, 124999875, 111111
(list;
graph;
refs;
listen;
history;
text;
internal format)



OFFSET

1,1


COMMENTS

"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)
Smith and Mikami put "x" whenever a decimal does not terminate. In the data, I put 1 instead of "x".
Murai Chuzen concludes that if 1 is divided by 9, 45, 3, 225, 18, 15, 1125, and 1, the results will have onedigit repetends; if 1 is divided by 99, 495, 33, 2475, 198, 165, 12375, and 11, the results will have twodigit repetends; if 1 is divided by 999, 4995, 333, 24975, 1998, 1665, 124875, and 111, the results will have threedigit repetends; etc.


REFERENCES

Murai Chuzen, Sampo Doshimon (Arithmetic for the Young), 1781.


LINKS

Table of n, a(n) for n=1..54.
David Eugene Smith and Yoshio Mikami, A history of Japanese mathematics, Open Court, 1914, reprinted by Dover, 2004, p. 176.


EXAMPLE

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.
99/1 = 99, so a(10) = 99; 99/2 = 49.5, so a(11) = 495.


CROSSREFS

Cf. A001913, A007732, A066799, A096688, A121090, A121341, A181431.
Sequence in context: A301397 A125679 A037207 * A096688 A181431 A249067
Adjacent sequences: A225485 A225486 A225487 * A225489 A225490 A225491


KEYWORD

base,sign


AUTHOR

Jonathan Sondow, May 10 2013


STATUS

approved



