A210770 -id:A210770 - cp's OEIS frontend

A210771 Inverse permutation to A210770.

1, 2, 4, 6, 3, 8, 5, 10, 12, 7, 14, 16, 18, 9, 20, 22, 11, 24, 26, 13, 28, 30, 15, 32, 17, 34, 36, 19, 38, 40, 21, 42, 44, 23, 46, 48, 25, 50, 52, 27, 54, 56, 29, 58, 60, 31, 62, 64, 66, 33, 68, 70, 35, 72, 74, 37, 76, 78, 39, 80, 82, 41, 84, 86, 43, 88, 90

Offset: 1

Reinhard Zumkeller, Mar 25 2012

nonn

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

import Data.List (elemIndex)
import Data.Maybe (fromJust)
a210771 n = fromJust (elemIndex n a210770_list) + 1

A064736 a(1)=1, a(2)=2; for n>0, a(2n+2) = smallest number missing from {a(1), ... ,a(2n)}, and a(2n+1) = a(2n)a(2n+2).

Original entry on oeis.org

1, 2, 6, 3, 12, 4, 20, 5, 35, 7, 56, 8, 72, 9, 90, 10, 110, 11, 143, 13, 182, 14, 210, 15, 240, 16, 272, 17, 306, 18, 342, 19, 399, 21, 462, 22, 506, 23, 552, 24, 600, 25, 650, 26, 702, 27, 756, 28, 812, 29, 870, 30, 930, 31, 992, 32, 1056, 33, 1122, 34, 1224, 36

Offset: 1

J. C. Lagarias (lagarias(AT)umich.edu), Oct 21 2001

Let c be the smallest positive constant such that for all permutations {a_n} of the positive integers, lim inf_{n -> infinity} gcd(a_n, a_{n+1})/n <= c. This sequence shows c >= 1/2.
The definition implies that if a(n) is prime then n is even. - N. J. A. Sloane, May 23 2017
a(2n) ~ n+1 ~ n has asymptotic density 1 and a(2n-1) ~ n(n+1) ~ n^2 has asymptotic density zero. - M. F. Hasler, May 23 2017

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Ray Chandler, Table of n, a(n) for n = 1..200000 (large file, 2.8 MB)
Ray Chandler, Table of n, a(n) for n = 1..2000000 (large gzipped file)
P. Erdős, R. Freud, and N. Hegyvári, Arithmetical properties of permutations of integers, Acta Mathematica Hungarica 41:1-2 (1983), pp. 169-176.
Dana G. Korssjoen, Biyao Li, Stefan Steinerberger, Raghavendra Tripathi, and Ruimin Zhang, Finding structure in sequences of real numbers via graph theory: a problem list, arXiv:2012.04625, Dec 08, 2020
Pierre Mazet, Eric Saias, Etude du graphe divisoriel 4, arXiv:1803.10073 [math.NT], 2018.
Index entries for sequences that are permutations of the natural numbers

A064745 gives inverse permutation.
Interleaving of A286290 and A286291. See also A286292, A286293.
Cf. A064764, A210770.

import Data.List (delete)
a064736 n = a064736_list !! (n-1)
a064736_list = 1 : 2 : f 1 2 [3..] where
   f u v (w:ws) = u' : w : f u' w (delete u' ws) where u' = v * w
-- Reinhard Zumkeller, Mar 23 2012

A064736 = {a[1]=1, a[2]=2}; a[n_] := a[n] = (an = If[OddQ[n], a[n-1]*a[n+1], First[ Complement[ Range[n], A064736]]]; AppendTo[A064736, an]; an); Table[a[n], {n, 1, 62}] (*Jean-François Alcover, Aug 07 2012 *)

More terms from Vladeta Jovovic, Oct 21 2001

Definition clarified by N. J. A. Sloane, May 23 2017

A055562 a(n) = least number greater than a(n-1) not the sum of an earlier pair of consecutive terms, a(0) = 2.

Original entry on oeis.org

2, 3, 4, 6, 8, 9, 11, 12, 13, 15, 16, 18, 19, 21, 22, 24, 26, 27, 29, 30, 32, 33, 35, 36, 38, 39, 41, 42, 44, 45, 47, 48, 49, 51, 52, 54, 55, 57, 58, 60, 61, 63, 64, 66, 67, 69, 70, 72, 73, 75, 76, 78, 79, 81, 82, 84, 85, 87, 88, 90, 91, 93, 94, 96, 98, 99, 101, 102, 104, 105

Offset: 0

Henry Bottomley, May 26 2000

nonn

			a(2) = 4 because a(1) = 3 and 4 <> a(0)+a(1);
a(3) = 6 because a(2) = 4 and 5 = a(0)+a(1) but 6 <> a(0)+a(1) and 6 <> a(1)+a(2).

Ivan Neretin, Table of n, a(n) for n = 0..10000
J.-P. Allouche and J. Shallit, The Ring of k-regular Sequences, II
J.-P. Allouche and J. Shallit, The ring of k-regular sequences, II, Theoret. Computer Sci., 307 (2003), 3-29.

Complement of A022441. See A001651 for a(0) = 1 and A055563 for a(0) = 3

a(n) = A022441(n) - a(n-1) for n > 0.
a(2n) = 3n + 1 + (floor(log_2 n) mod 2), n >= 1; a(2n+1) = 3n+3, n >= 0. - Jeffrey Shallit, Jun 08 2000
a(n) = A210770(2*n+2). - Reinhard Zumkeller, Mar 25 2012

A022441 a(n) = c(n) + c(n-1) where c (A055562) is the sequence of numbers not in a.

Original entry on oeis.org

1, 5, 7, 10, 14, 17, 20, 23, 25, 28, 31, 34, 37, 40, 43, 46, 50, 53, 56, 59, 62, 65, 68, 71, 74, 77, 80, 83, 86, 89, 92, 95, 97, 100, 103, 106, 109, 112, 115, 118, 121, 124, 127, 130, 133, 136, 139, 142, 145, 148, 151, 154, 157, 160, 163, 166, 169, 172, 175, 178

Offset: 0

Clark Kimberling

Ivan Neretin, Table of n, a(n) for n = 0..10000
J-P. Bode, H. Harborth, C. Kimberling, Complementary Fibonacci sequences, Fibonacci Quarterly 45 (2007), 254-264.
R. Stephan, Some divide-and-conquer sequences ...
R. Stephan, Table of generating functions

Cf. A055562 (complement), A022424.

[1] cat [3*n + 2 - (Floor((Log(n)/Log(2))) mod 2): n in [0..10]]; // G. C. Greubel, Mar 08 2018

A022441 := n-> `if`(n=0, 1, 3*n + 2 - (ilog2(n) mod 2)):
seq(A022441(n), n= 0..59);

Fold[Append[#1, Plus @@ Complement[Range[Max@#1 + 3], #1][[{#2 + 1, #2 + 2}]]] &, {1, 5}, Range[58]] (* Ivan Neretin, Mar 30 2017 *)
Table[If[n==0,1, 3*n+2 - Mod[Floor[Log[n]/Log[2]], 2]], {n,0,30}] (* G. C. Greubel, Mar 08 2018 *)

for(n=0,30, print1(if(n==0,1, 3*n+2 - (floor(log(n)/log(2))%2)), ", ")) \\ G. C. Greubel, Mar 08 2018

a(n) + a(n-1) = 3n + 2 - (floor(log_2 n) mod 2) for n >= 1. - Jeffrey Shallit, Jun 08 2000
For n>0, a(n) = b(n) with b(0)=0, b(2n) = -b(n)+9n+3, b(2n+1) = -b(n)+9n+6-[n==0]. - Ralf Stephan, Oct 24 2003
a(n) = A210770(2*n+1). - Reinhard Zumkeller, Mar 25 2012

More terms from Winston C. Yang (winston(AT)cs.wisc.edu), Aug 23 2000
Term a(16)=50 fixed by Ivan Neretin, Mar 30 2017
Updated by Clark Kimberling, Feb 19 2018

cp's OEIS Frontend

A210771 Inverse permutation to A210770.

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A064736 a(1)=1, a(2)=2; for n>0, a(2n+2) = smallest number missing from {a(1), ... ,a(2n)}, and a(2n+1) = a(2n)a(2n+2).

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A055562 a(n) = least number greater than a(n-1) not the sum of an earlier pair of consecutive terms, a(0) = 2.

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A022441 a(n) = c(n) + c(n-1) where c (A055562) is the sequence of numbers not in a.

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cp's OEIS Frontend

A210771 Inverse permutation to A210770.

Views

Author

Keywords

Links

Programs

Haskell

A064736 a(1)=1, a(2)=2; for n>0, a(2*n+2) = smallest number missing from {a(1), ... ,a(2*n)}, and a(2*n+1) = a(2*n)*a(2*n+2).

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Author

Keywords

Comments

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Crossrefs

Programs

Haskell

Mathematica

Extensions

A055562 a(n) = least number greater than a(n-1) not the sum of an earlier pair of consecutive terms, a(0) = 2.

Views

Author

Keywords

Examples

Links

Crossrefs

Formula

A022441 a(n) = c(n) + c(n-1) where c (A055562) is the sequence of numbers not in a.

Views

Author

Keywords

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

Formula

Extensions

A064736 a(1)=1, a(2)=2; for n>0, a(2n+2) = smallest number missing from {a(1), ... ,a(2n)}, and a(2n+1) = a(2n)a(2n+2).