cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A113220 Inverse of A113218.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 10, 13, 12, 15, 14, 17, 16, 19, 21, 18, 23, 20, 25, 22, 27, 24, 29, 31, 26, 33, 28, 35, 30, 37, 32, 39, 41, 34, 43, 36, 45, 38, 47, 40, 49, 51, 42, 53, 44, 55, 46, 57, 48, 59, 61, 50, 63, 52, 65, 54, 67, 56, 69, 71, 58, 73, 60, 75, 62, 77, 64, 79
Offset: 0

Views

Author

Reinhard Zumkeller, Oct 18 2005

Keywords

Comments

Permutation of the natural numbers.

Links

A113219 A113218(A113218(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 22, 20, 24, 18, 26, 19, 29, 21, 27, 23, 28, 25, 30, 31, 32, 33, 42, 35, 44, 38, 47, 40, 49, 34, 43, 36, 45, 37, 46, 39, 48, 41, 62, 51, 65, 53, 67, 56, 69, 58, 59, 60, 61, 50, 63, 52, 64, 54, 83, 55, 85, 57, 87, 71, 89
Offset: 0

Views

Author

Reinhard Zumkeller, Oct 18 2005

Keywords

Comments

Inverse integer permutation of A113221;
A113220(a(n)) = a(A113220(n)) = A113218(n).

Links

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

Original entry on oeis.org

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

Views

Author

N. J. A. Sloane

Keywords

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[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

Examples

			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.

References

  • Martin Gardner, Mathematics, Magic and Mystery, 1956.

Links

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).

Programs

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

A113217 Parity of decimal digital root of n.

Original entry on oeis.org

0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 0, 1
Offset: 0

Views

Author

Reinhard Zumkeller, Oct 18 2005

Keywords

Comments

Except for the first element, the sequence is periodic (with a period of length 9). The sequence corresponds to that produced by a prescribed set of bitwise operations. The (sub)sequence is produced starting from input pairs (0,1),(1,1),(1,0). For example, (0,1) acted on (in succession) by [and,xor,or,xor,or,and,or,and,xor], with the same operation set then repeated. For clarity, the example is AND(0,1) is 0. XOR(1,0) is 1. OR(0,1) is 1. XOR(1,1) is 0. OR(1,0) is 1. AND(0,1) is 0. OR(1,0) is 1. AND(0,1) is 0. XOR(1,0) is 1. Repeat. The analysis was done using Gnumeric's built-in functions. In this example, the inputs align to n=2,3, and the operation results to the next 7 elements. The (3) starting input pairs mentioned begin at bitwise operator positions 1,2 and 5. - Bill McEachen, May 24 2014

Links

Crossrefs

Cf. A000035, A010888, A113218.

Programs

  • Mathematica
    Table[Mod[ResourceFunction["AdditiveDigitalRoot"][n],2],{n,0,104}] (* James C. McMahon, Jun 19 2024 *)

Formula

a(n) = A010888(n) mod 2.
a(n) = if n mod 9 = 1 then 1 else 1 - a(n-1), a(0)=0.
a(n) = A000035(A010888(n)). - Omar E. Pol, Oct 28 2013
a(n) = (1+(-1)^floor(8*n/9))/2 for n>0. - Wesley Ivan Hurt, Apr 27 2020

A113221 A113220(A113220(n)).

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 21, 23, 19, 25, 18, 27, 20, 29, 22, 26, 28, 24, 30, 31, 32, 33, 41, 35, 43, 45, 37, 47, 39, 49, 34, 42, 36, 44, 46, 38, 48, 40, 61, 51, 63, 53, 65, 67, 55, 69, 57, 58, 59, 60, 50, 62, 64, 52, 81, 54, 83, 56, 85, 71, 87
Offset: 0

Views

Author

Reinhard Zumkeller, Oct 18 2005

Keywords

Comments

Inverse integer permutation of T000004;
A113218(a(n)) = a(A113218(n)) = A113220(n).

Links

Showing 1-5 of 5 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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