A113217 -id:A113217 - cp's OEIS frontend

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

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

Offset: 0

N. J. A. Sloane

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[123456789] (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

			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.

Martin Gardner, Mathematics, Magic and Mystery, 1956.

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 Colombian or self numbers and related sequences
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,1).

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).

a010888 = until (< 10) a007953
-- Reinhard Zumkeller, Oct 17 2011, May 12 2011

[n eq 0 select 0 else 1+(n-1) mod 9: n in [0..110]]; // Bruno Berselli, Mar 18 2016

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

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 *)

A010888(n)=if(n,(n-1)%9+1) \\ M. F. Hasler, Jan 04 2011

def A010888(n):
    return 1 + (n - 1) % 9 if n else 0 # Chai Wah Wu, Aug 23 2014, Apr 23 2023

0 :: List.fill(10)(1 to 9).flatten // Alonso del Arte, Feb 01 2020

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) = n - 9*floor((n-1)/9), for n > 0. - José de Jesús Camacho Medina, Nov 10 2014

A113218 Lexicographically earliest permutation of the natural numbers with alternating parities of digital roots in decimal representation.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 13, 12, 15, 14, 17, 16, 20, 18, 22, 19, 24, 21, 26, 23, 29, 25, 31, 27, 33, 28, 35, 30, 38, 32, 40, 34, 42, 36, 44, 37, 47, 39, 49, 41, 51, 43, 53, 45, 56, 46, 58, 48, 60, 50, 62, 52, 65, 54, 67, 55, 69, 57, 71, 59, 74, 61, 76, 63, 78, 64, 80

Offset: 0

Reinhard Zumkeller, Oct 18 2005

inverse: A113220; A113219 = a(a(n));

A113217(a(n)) = A000035(n).

Index entries for sequences that are permutations of the natural numbers
Eric Weisstein's World of Mathematics, Digital Root

Cf. A010888.

A298638 Numbers k such that the digital sum of k and the digital root of k have opposite parity.

Original entry on oeis.org

19, 28, 29, 37, 38, 39, 46, 47, 48, 49, 55, 56, 57, 58, 59, 64, 65, 66, 67, 68, 69, 73, 74, 75, 76, 77, 78, 79, 82, 83, 84, 85, 86, 87, 88, 89, 91, 92, 93, 94, 95, 96, 97, 98, 99, 109, 118, 119, 127, 128, 129, 136, 137, 138, 139, 145, 146, 147, 148, 149, 154, 155

Offset: 1

J. Stauduhar, Jan 23 2018

Numbers k such that A113217(k) <> A179081(k).
Complement of A298639.
Agrees with A291884 until a(46): a(46) = 109 is not in that sequence.

J. Stauduhar, Table of n, a(n) for n = 1..10000

Cf. A007953, A010888, A113217, A179081.

Cf. A298639, A291884.

Select[Range[145], EvenQ@ Total@ IntegerDigits@ # != EvenQ@ NestWhile[Total@ IntegerDigits@ # &, #, # > 9 &] &] (* Michael De Vlieger, Feb 03 2018 *)

isok(n) = sumdigits(n) % 2 != if (n, ((n-1)%9+1) % 2, 0); \\ Michel Marcus, Mar 01 2018

#Digital sum of n.
def ds(n):
  if n < 10:
    return n
  return n % 10 + ds(n//10)
def A298638(term_count):
  seq = []
  m = 0
  n = 1
  while n <= term_count:
    s = ds(m)
    r = ((m - 1) % 9) + 1 if m else 0
    if s % 2 != r % 2:
      seq.append(m)
      n += 1
    m += 1
  return seq
print(A298638(100))

A298639 Numbers k such that the digital sum of k and the digital root of k have the same parity.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 20, 21, 22, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 35, 36, 40, 41, 42, 43, 44, 45, 50, 51, 52, 53, 54, 60, 61, 62, 63, 70, 71, 72, 80, 81, 90, 100, 101, 102, 103, 104, 105, 106, 107, 108, 110, 111, 112, 113, 114

Offset: 1

J. Stauduhar, Jan 26 2018

Numbers k such that A113217(k) = A179081(k).
Complement of A298638.
Agrees with A039691 until a(65): A039691(65) = 109 is not in this sequence.

J. Stauduhar, Table of n, a(n) for n = 1..10000

Cf. A007953, A010888, A113217, A179081, A298638, A039691.

fQ[n_] := Mod[Plus @@ IntegerDigits@n, 2] == Mod[Mod[n -1, 9] +1, 2]; fQ[0] = True; Select[ Range[0, 104], fQ] (* Robert G. Wilson v, Jan 26 2018 *)

dr(n)=if(n, (n-1)%9+1);
isok(n) = (sumdigits(n) % 2) == (dr(n) % 2); \\ Michel Marcus, Jan 26 2018

is(n)=bittest(sumdigits(n)-(n-1)%9,0)||!n \\ M. F. Hasler, Jan 26 2018

#Digital sum of n.
def ds(n):
  if n < 10:
    return n
  return n % 10 + ds(n//10)
def A298639(term_count):
  seq = []
  m = 0
  n = 1
  while n <= term_count:
    s = ds(m)
    r = ((m - 1) % 9) + 1 if m else 0
    if s % 2 == r % 2:
      seq.append(m)
      n += 1
    m += 1
  return seq
print(A298639(100))

cp's OEIS Frontend

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

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A113218 Lexicographically earliest permutation of the natural numbers with alternating parities of digital roots in decimal representation.

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A298638 Numbers k such that the digital sum of k and the digital root of k have opposite parity.

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A298639 Numbers k such that the digital sum of k and the digital root of k have the same parity.

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Author

Keywords

Comments

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Crossrefs

Programs

Mathematica

PARI

PARI

Python