

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
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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



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



KEYWORD

base,sign


AUTHOR



STATUS

approved



