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 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 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. REFERENCES Murai Chuzen, Sampo Doshi-mon (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

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Last modified May 19 07:05 EDT 2024. Contains 372666 sequences. (Running on oeis4.)