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A010888 Digital root of n (repeatedly add the digits of n until a single digit is reached). 225

%I

%S 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,

%T 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,

%U 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

%N Digital root of n (repeatedly add the digits of n until a single digit is reached).

%C This is sometimes also called the additive digital root of n.

%C n mod 9 (A010878) is a very similar sequence.

%C Partial sums are given by A130487(n-1) + n (for n>0). - _Hieronymus Fischer_, Jun 08 2007

%C Decimal expansion of 13717421/111111111 is 0.123456789123456789123456789... with period 9. - _Eric Desbiaux_, May 19 2008

%C Decimal expansion of 13717421 / 1111111110 = 0.0[123456789] (periodic) - _Daniel Forgues_, Feb 27 2017

%C a(A005117(n)) < 9. - _Reinhard Zumkeller_, Mar 30 2010

%C 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

%D Martin Gardner, Mathematics, Magic and Mystery, 1956.

%H N. J. A. Sloane, <a href="/A010888/b010888.txt">Table of n, a(n) for n = 0..10000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Digitaddition.html">Digitaddition</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DigitalRoot.html">Digital Root</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Vedic_square">Vedic square</a>

%H <a href="/index/Coi#Colombian">Index entries for Colombian or self numbers and related sequences</a>

%H <a href="/index/Rec#order_09">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 0, 0, 0, 1).

%F If n=0 then a(n) = 0; otherwise a(n) = (n reduced mod 9), but if the answer is 0 change it to 9.

%F Equivalently, if n=0 then a(n) = 0, otherwise a(n) = (n-1 reduced mod 9) + 1.

%F If the initial 0 term is ignored, the sequence is periodic with period 9.

%F From _Hieronymus Fischer_, Jun 08 2007: (Start)

%F a(n) = A010878(n-1) + 1 (for n>0).

%F 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)

%F 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

%F a(n) = n-9*floor((n-1)/9), for n>0. - _José de Jesús Camacho Medina_, Nov 10 2014

%e 37 -> 10 -> 1, so a(37)=1.

%p A010888 := n->if n=0 then 0 else ((n-1) mod 9) + 1; fi; # _N. J. A. Sloane_, Feb 20 2013

%t Join[{0}, Array[Mod[ # - 1, 9] + 1 &, 104]] (* _Robert G. Wilson v_, Jan 04 2006 *)

%t 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 *)

%t 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 *)

%t Table[FixedPoint[Total[IntegerDigits[#, 10]] &, n], {n, 0, 104}] (* _IWABUCHI Yu(u)ki_, Jun 03 2016 *)

%o (PARI) A010888(n)=if(n,(n-1)%9+1) \\ _M. F. Hasler_, Jan 04 2011

%o (Haskell)

%o a010888 = until (< 10) a007953

%o -- _Reinhard Zumkeller_, Oct 17 2011, May 12 2011

%o (Python)

%o def A010888(n):

%o ....return 1+ (n-1) % 9 # _Chai Wah Wu_, Aug 23 2014

%o (MAGMA) [n eq 0 select 0 else 1+(n-1) mod 9: n in [0..110]]; // _Bruno Berselli_, Mar 18 2016

%Y 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).

%K nonn,easy,nice,base

%O 0,3

%A _N. J. A. Sloane_

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Last modified October 22 09:57 EDT 2018. Contains 316433 sequences. (Running on oeis4.)