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 A010888 Digital root of n (repeatedly add the digits of n until a single digit is reached). 234
 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5, 6, 7, 8, 9, 1, 2, 3, 4, 5 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 COMMENTS This is sometimes also called the additive digital root of n. n mod 9 (A010878) is a very similar sequence. Partial sums are given by A130487(n-1) + n (for n > 0). - Hieronymus Fischer, Jun 08 2007 Decimal expansion of 13717421/111111111 is 0.123456789123456789123456789... with period 9. - Eric Desbiaux, May 19 2008 Decimal expansion of 13717421 / 1111111110 = 0.0 (periodic) - Daniel Forgues, Feb 27 2017 a(A005117(n)) < 9. - Reinhard Zumkeller, Mar 30 2010 My friend Jahangeer Kholdi has found that 19 is the smallest prime p such that for each number n, a(p*n) = a(n). In fact we have: a(m*n) = a(a(m)*a(n)) so all numbers with digital root 1 (numbers of the form 9k + 1) have this property. See comment lines of A017173. Also we have a(m+n) = a(a(m) + a(n)). - Farideh Firoozbakht, Jul 23 2010 REFERENCES Martin Gardner, Mathematics, Magic and Mystery, 1956. LINKS N. J. A. Sloane, Table of n, a(n) for n = 0..10000 Eric Weisstein's World of Mathematics, Digitaddition Eric Weisstein's World of Mathematics, Digital Root Wikipedia, Vedic square Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1). FORMULA If n = 0 then a(n) = 0; otherwise a(n) = (n reduced mod 9), but if the answer is 0 change it to 9. Equivalently, if n = 0 then a(n) = 0, otherwise a(n) = (n - 1 reduced mod 9) + 1. If the initial 0 term is ignored, the sequence is periodic with period 9. From Hieronymus Fischer, Jun 08 2007: (Start) a(n) = A010878(n-1) + 1 (for n > 0). G.f.: g(x) = x*(Sum_{k = 0..8}(k+1)*x^k)/(1 - x^9). Also: g(x) = x(9x^10 - 10x^9 + 1)/((1 - x^9)(1 - x)^2). (End) a(n) = 1 + ((n+8) mod 9) - 9*(1 - (((n+1)! + 1) mod (n+1))). a(n) = 1 + ((n+8) mod 9) - 9*A000007. - Paolo P. Lava, Jun 20 2007 a(n) = n - 9*floor((n-1)/9), for n > 0. - José de Jesús Camacho Medina, Nov 10 2014 EXAMPLE The digits of 37 are 3 and 7, and 3 + 7 = 10. And the digits of 10 are 1 and 0, and 1 + 0 = 1, so a(37) = 1. MAPLE A010888 := n->if n=0 then 0 else ((n-1) mod 9) + 1; fi; # N. J. A. Sloane, Feb 20 2013 MATHEMATICA Join[{0}, Array[Mod[ # - 1, 9] + 1 &, 104]] (* Robert G. Wilson v, Jan 04 2006 *) Join[Range[0, 1], Table[n - 9 Floor[(n - 1) / 9], {n, 2, 100}]] (* José de Jesús Camacho Medina, Nov 10 2014 *) (* Corrected by Vincenzo Librandi, Nov 11 2014 *) Join[{0}, LinearRecurrence[{0, 0, 0, 0, 0, 0, 0, 0, 1}, {1, 2, 3, 4, 5, 6, 7, 8, 9}, 104]] (* Ray Chandler, Aug 26 2015 *) Table[FixedPoint[Total[IntegerDigits[#, 10]] &, n], {n, 0, 104}] (* IWABUCHI Yu(u)ki, Jun 03 2016 *) PROG (PARI) A010888(n)=if(n, (n-1)%9+1) \\ M. F. Hasler, Jan 04 2011 (Haskell) a010888 = until (< 10) a007953 -- Reinhard Zumkeller, Oct 17 2011, May 12 2011 (Python) def A010888(n):     return 1 + (n - 1) % 9 # Chai Wah Wu, Aug 23 2014 (MAGMA) [n eq 0 select 0 else 1+(n-1) mod 9: n in [0..110]]; // Bruno Berselli, Mar 18 2016 (Scala) 0 :: List.fill(10)(1 to 9).flatten // Alonso del Arte, Feb 01 2020 CROSSREFS Cf. A007953, A007954, A031347, A113217, A113218, A010878 (n mod 9), A010872, A010873, A010874, A010875, A010876, A010877, A010879, A004526, A002264, A002265, A002266, A017173, A031286 (additive persistence of n), (multiplicative digital root of n), A031346 (multiplicative persistence of n). Sequence in context: A285093 A053837 A128244 * A177274 A131650 A033930 Adjacent sequences:  A010885 A010886 A010887 * A010889 A010890 A010891 KEYWORD nonn,easy,nice,base AUTHOR STATUS approved

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Last modified September 18 07:39 EDT 2020. Contains 337166 sequences. (Running on oeis4.)