A006368 -id:A006368 - cp's OEIS frontend

A223083 Trajectory of 64 under the map n-> A006369(n).

64, 85, 113, 151, 201, 134, 179, 239, 319, 425, 567, 378, 252, 168, 112, 149, 199, 265, 353, 471, 314, 419, 559, 745, 993, 662, 883, 1177, 1569, 1046, 1395, 930, 620, 827, 1103, 1471, 1961, 2615, 3487, 4649, 6199, 8265, 5510, 7347, 4898, 6531, 4354, 5805

Offset: 1

N. J. A. Sloane, Mar 22 2013

nonn

It is conjectured that this trajectory does not close on itself.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.

Cf. A006369, A006368, A182205.

Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589, A185590.

f:=proc(N) if N mod 3 = 0 then 2*(N/3); elif N mod 3 = 2 then 4*((N+1)/3)-1; else 4*((N+2)/3)-3; fi; end;
t1:=[64];
for n from 1 to 100 do t1:=[op(t1),f(t1[nops(t1)])]; od:
t1;

t = {64}; While[n = t[[-1]]; s = Switch[Mod[n, 3], 0, 2*n/3, 1, (4*n - 1)/3, 2, (4*n + 1)/3]; Length[t] < 100 && ! MemberQ[t, s], AppendTo[t, s]]; t (* T. D. Noe, Mar 22 2013 *)
SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n - 1)/3, , (4n + 1)/3]}, {64}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)

A265667 Permutation of nonnegative integers: a(n) = n + floor(n/3)*(-1)^(n mod 3).

Original entry on oeis.org

0, 1, 2, 4, 3, 6, 8, 5, 10, 12, 7, 14, 16, 9, 18, 20, 11, 22, 24, 13, 26, 28, 15, 30, 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, 33, 66, 68, 35, 70, 72, 37, 74, 76, 39, 78, 80, 41, 82, 84, 43, 86, 88, 45

Offset: 0

Bruno Berselli, Dec 12 2015 - based on an idea by Paul Curtz

The inverse permutation is given by P(n) = A006368(n-1) + 1, for n >= 1, and P(0) = 0. - Wolfdieter Lang, Sep 21 2021

This permutation is given by A006369(n-1) + 1, with A006369(-1) = -1. Observed by Kevin Ryde. - Wolfdieter Lang, Sep 22 2021

			-------------------------------------------------------------------------
0, 1, 2, 3,  4, 5, 6,  7,  8,  9, 10, 11, 12, 13, 14, 15, 16, 17, 18, ...
+  +  +  +   +  +  +   +   +   +   +   +   +   +   +   +   +   +   +
0, 0, 0, 1, -1, 1, 2, -2,  2,  3, -3,  3,  4, -4,  4,  5, -5,  5,  6, ...
-------------------------------------------------------------------------
0, 1, 2, 4,  3, 6, 8,  5, 10, 12,  7, 14, 16,  9, 18, 20, 11, 22, 24, ...
-------------------------------------------------------------------------

Bruno Berselli, Table of n, a(n) for n = 0..1000
Peter Lynch and Michael Mackey, Parity and Partition of the Rational Numbers, arXiv:2205.00565 [math.NT], 2022. See set F p. 4.
Index entries for linear recurrences with constant coefficients, signature (0,0,2,0,0,-1).
Index entries for permutations of the positive (or nonnegative) integers.

Cf. A001477, A006368, A006369, A008738, A032793.
Cf. A064455: n+floor(n/2)*(-1)^(n mod 2).
Cf. A265888: n+floor(n/4)*(-1)^(n mod 4).
Cf. A265734: n+floor(n/5)*(-1)^(n mod 5).

[n+Floor(n/3)*(-1)^(n mod 3): n in [0..70]];

Table[n + Floor[n/3] (-1)^Mod[n, 3], {n, 0, 70}]

[n+floor(n/3)*(-1)^mod(n,3) for n in (0..70)]

G.f.: x*(1 + 2*x + 4*x^2 + x^3 + 2*x^4) / ((1 - x)^2*(1 + x + x^2)^2).
a(n) = 2*a(n-3) - a(n-6).
a(3*k)   = 4*k;
a(3*k+1) = 2*k+1, hence a(3*k+1) = a(3*k)/2 + 1;
a(3*k+2) = 4*k+2, hence a(3*k+2) = 2*a(3*k+1) = a(3*k) + 2.
Sum_{i=0..n} a(i) = A008738(A032793(n+1)).
Sum_{n>=1} (-1)^(n+1)/a(n) = Pi/8 + log(2)/4. - Amiram Eldar, Mar 30 2023

A131743 Period 4: repeat [0, 1, 0, 2].

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0, 1, 0, 2, 0

Offset: 0

Paul Curtz, Sep 20 2007

Least positive integer k such that n^k == 1 (mod 4), or 0 if GCD(n,4) > 1. - Bruno Berselli, Mar 22 2016

Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A006368.

&cat [[0, 1, 0, 2]^^30]; // Bruno Berselli, Mar 22 2016

seq(op([0, 1, 0, 2]), n=0..50); # Wesley Ivan Hurt, Jul 09 2016

PadRight[{}, 106, {0, 1, 0, 2}] (* Harvey P. Dale, Apr 06 2012 *)
CoefficientList[Series[x (1 + 2 x^2)/((1 - x) (x + 1) (x^2 + 1)), {x, 0, 120}], x] (* Michael De Vlieger, Jul 09 2016 *)

x='x+O('x^200); concat(0, Vec(-x*(1+2*x^2)/((x-1)*(x+1)*(x^2+1)))) \\ Altug Alkan, Mar 22 2016

a(n)=n%2*(n%4+1)/2 \\ Charles R Greathouse IV, Mar 22 2016

def A131743(n): return (0,1,0,2)[n&3] # Chai Wah Wu, Jul 29 2023

G.f.: x*(1+2*x^2)/ ((1-x)*(x+1)*(x^2+1)). - R. J. Mathar, Nov 15 2007
a(n) = 3/4-1/2*sin(1/2*Pi*n)+3/4*(-1)^(1+n). - R. J. Mathar, Nov 15 2007
a(n) = Fibonacci(2*n) mod 3. - Gary Detlefs Feb 13 2011
a(n) == A006368(n) (mod 3). - Philippe Deléham, Oct 24 2011
a(n) = a(n-4) for n>3. - Wesley Ivan Hurt, Jul 09 2016

A168222 a(n) = A006369(A006369(n)).

Original entry on oeis.org

0, 1, 2, 3, 7, 9, 5, 6, 15, 4, 17, 10, 11, 23, 25, 13, 14, 31, 8, 33, 18, 19, 39, 41, 21, 22, 47, 12, 49, 26, 27, 55, 57, 29, 30, 63, 16, 65, 34, 35, 71, 73, 37, 38, 79, 20, 81, 42, 43, 87, 89, 45, 46, 95, 24, 97, 50, 51, 103, 105, 53, 54, 111, 28, 113, 58, 59, 119, 121, 61, 62

Offset: 0

Reinhard Zumkeller, Nov 20 2009

nonn

Inverse integer permutation to A168221;

a(A006368(n)) = A006369(n).

Table[Nest[If[Divisible[#,3],2#/3,Round[4#/3]]&,n,2],{n,0,100}] (* Paolo Xausa, Dec 15 2023 *)

Conjectures from Colin Barker, Aug 15 2019: (Start)
G.f.: x*(1 + 2*x + 3*x^2 + 7*x^3 + 9*x^4 + 5*x^5 + 6*x^6 + 15*x^7 + 4*x^8 + 15*x^9 + 6*x^10 + 5*x^11 + 9*x^12 + 7*x^13 + 3*x^14 + 2*x^15 + x^16) / ((1 - x)^2*(1 + x + x^2)^2*(1 + x^3 + x^6)^2).
a(n) = 2*a(n-9) - a(n-18) for n>17.
(End)

A349378 a(n) = A349376(n) + A349377(n).

Original entry on oeis.org

2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, -6, 0, 0, 0, 10, 9, 0, 0, -8, 0, -30, 0, -4, 0, 0, 24, 27, 0, 0, 0, 34, 0, -40, 0, -14, -6, 0, 0, -34, 16, -66, 0, -14, 0, 10, 42, 44, 0, 0, 0, 166, 0, 0, -8, 11, 42, -70, 0, -20, 0, -172, 0, -70, 0, 0, -42, -22, 56, -70, 0, -103, 1, 0, 0, 216, 60, 0, 0, 78

Offset: 1

Antti Karttunen, Nov 17 2021

sign

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A006368, A006369, A349376, A349377.

Cf. also A349352, A349369.

A349378(n) = (A349376(n)+A349377(n)); \\ Needs also code from A349376 and A349377.

a(1) = 2, and for n >1, a(n) = -Sum_{d|n, 1A349376(d) * A349377(n/d). [As the sequences are Dirichlet inverses of each other]

A178414 Least odd number in the Collatz (3x+1) preimage of odd numbers not a multiple of 3.

Original entry on oeis.org

1, 3, 9, 7, 17, 11, 25, 15, 33, 19, 41, 23, 49, 27, 57, 31, 65, 35, 73, 39, 81, 43, 89, 47, 97, 51, 105, 55, 113, 59, 121, 63, 129, 67, 137, 71, 145, 75, 153, 79, 161, 83, 169, 87, 177, 91, 185, 95, 193, 99, 201, 103, 209, 107, 217, 111, 225, 115, 233, 119, 241, 123, 249

Offset: 1

T. D. Noe, May 28 2010

The odd non-multiples of 3 are 1, 5, 7, 11,... (A007310). The odd multiples of 3 have no odd numbers their Collatz pre-image. The next odd number in the Collatz iteration of a(2n) is 6n-1. The next odd number in the Collatz iteration of a(2n+1) is 6n+1. For each non-multiple of 3, there are an infinite number of odd numbers in its Collatz pre-image. For example:
Odd pre-images of 1: 1, 5, 21, 85, 341,... (A002450)
Odd pre-images of 5: 3, 13, 53, 213, 853,... (A072197)
Odd pre-images of 7: 9, 37, 149, 597, 2389,...
Odd pre-images of 11: 7, 29, 117, 469, 1877,...(A072261)
In each case, the pre-image sequence is t(k+1) = 4*t(k) + 1 with t(0)=a(n). The array of pre-images is in A178415.
a(n) = A047529(P(n)), with the permutation P(n) = A006368(n-1) + 1, for n >= 1. This shows that this sequence gives the numbers {1, 3, 7} (mod 8) uniquely. - Wolfdieter Lang, Sep 21 2021

Matthew House, Table of n, a(n) for n = 1..10000
Index entries for linear recurrences with constant coefficients, signature (0,2,0,-1).

Cf. A002450, A006368, A007310, A022998, A047529, A072197, A072261, A114752.

Riffle[1+8*Range[0,50], 3+4*Range[0,50]]

a(n) = (n - 1)*(3 - (-1)^n) + 1. [Bogart B. Strauss, Sep 20 2013, adapted to the offset by Matthew House, Feb 14 2017]
From Matthew House, Feb 14 2017: (Start)
G.f.: x*(1 + 3*x + 7*x^2 + x^3)/((1 - x)^2*(1 + x)^2).
a(n) = 2*a(n-2) - a(n-4). (End)
From Philippe Deléham, Nov 06 2023: (Start)
a(2*n) = 4*n-1, a(2*n+1) = 8*n+1.
a(n) = 2*A022998(n-1)+1.
a(n) = 2*A114752(n)-1. (End)

A006373 Numbers of terms in expressions for coefficients of Euler-Lagrange tensors in terms of Riemann-Christoffel curvature tensor and two of its contractions (viz., the Ricci curvature tensor and the Riemann curvature scalar) for n-dimensional differentiable manifolds having a general linear connection.

Original entry on oeis.org

1, 2, 7, 26, 115, 596

Offset: 0

C. C. Briggs (ccb104(AT)vm.cac.psu.edu)

The six known terms of this sequence coincide with the first six terms of A167551. - Johannes W. Meijer, Nov 12 2009

C. C. Briggs, A General Expression for the Quintic Lovelock Tensor, arXiv:gr-qc/9607033, 1996-1997.
C. C. Briggs, A General Expression for the Quartic Lovelock Tensor, arXiv:gr-qc/9703074, 1997.
S. Schreiber & N. J. A. Sloane, Correspondence, 1980

Cf. A006372, A030429.

Cf. A167551.

A330530 Lexicographically earliest sequence of distinct positive integers such that the product of two consecutive terms is always divisible by 4.

Original entry on oeis.org

1, 4, 2, 6, 8, 3, 12, 5, 16, 7, 20, 9, 24, 10, 14, 18, 22, 26, 28, 11, 32, 13, 36, 15, 40, 17, 44, 19, 48, 21, 52, 23, 56, 25, 60, 27, 64, 29, 68, 30, 34, 38, 42, 46, 50, 54, 58, 62, 66, 70, 72, 31, 76, 33, 80, 35, 84, 37, 88, 39, 92, 41, 96, 43, 100, 45, 104

Offset: 1

Rémy Sigrist, Dec 17 2019

For any k > 0, let f_k be the lexicographically earliest sequence of distinct positive integers such that the product of two consecutive terms is always divisible by k:
- in particular:
     f_1 = f_2 = A000027,
     f_3 = A006368,
     f_4 = a (this sequence),
     f_6 = A330531,
- f_k is a permutation of the natural numbers,
- f_k(1) = 1, f_k(2) = max(2, k),
- if k is prime, then f_k corresponds to the integers that are not multiple of k interspersed with the integers that are multiple of k.
Apparently:
- for m > 0, the m-th run of consecutive terms such that gcd(a(n), 4) = 2 has A153893(m+1) terms,
- for m > 1, the m-th run of consecutive terms such that gcd(a(n), 4) = 1 or 4 has A068156(m+1) terms.

			The first terms, alongside their product with the next term, are:
  n   a(n)  a(n)*a(n+1)
  --  ----  -----------
   1     1            4
   2     4            8
   3     2           12
   4     6           48
   5     8           24
   6     3           36
   7    12           60
   8     5           80
   9    16          112
  10     7          140

Rémy Sigrist, Table of n, a(n) for n = 1..10000
Rémy Sigrist, Colored scatterplot of the first 10000 terms
Index entries for sequences that are permutations of the natural numbers

Cf. A006368, A068156, A153893, A330531 (f_6), A330576 (inverse).

s=0; v=1; for (n=1, 10 000, print (n " " v); s+=2^v; for (w=1, oo, if (!bittest(s,w) && (v*w)%4==0, v=w; break)))

A342131 a(n) = n/2 + floor(n/4) if n is even, otherwise (3*n+1)/2.

Original entry on oeis.org

0, 2, 1, 5, 3, 8, 4, 11, 6, 14, 7, 17, 9, 20, 10, 23, 12, 26, 13, 29, 15, 32, 16, 35, 18, 38, 19, 41, 21, 44, 22, 47, 24, 50, 25, 53, 27, 56, 28, 59, 30, 62, 31, 65, 33, 68, 34, 71, 36, 74, 37, 77, 39, 80, 40, 83, 42, 86, 43, 89, 45, 92, 46, 95, 48, 98, 49, 101, 51, 104

Offset: 0

Thomas Scheuerle, Mar 01 2021

A permutation of the nonnegative integers related to the Collatz function (A014682).

Interspersion of A032766 and A016789. - Michel Marcus, Mar 04 2021

Cf. A014682, A006368.

Cf. A032766, A016789.

function [a] = A342131(max_n)
    for n = 1:max_n
        m = n-1;
        if floor(m/2) == m/2
            a(n) = (m/2)+floor(m/4);
        else
            a(n) = (m*3+1)/2;
        end
    end
end

&cat [[3*k,6*k+2,3*k+1,6*k+5]: k in [0..20]] // Bruno Berselli, Mar 05 2021

a[n_] := If[EvenQ[n], n/2 + Floor[n/4], (3*n + 1)/2]; Array[a, 100, 0] (* Amiram Eldar, Mar 03 2021 *)
Table[(12 n + 4 - (3 n + 3 - (-1)^(n/2)) (1 + (-1)^n))/8, {n, 0, 70}] (* Bruno Berselli, Mar 05 2021 *)

a(n) = if (n%2, (3*n+1)/2, n/2 + n\4); \\ Michel Marcus, Mar 04 2021

def A342131(n): return (3*n+1)//2 if n % 2 else n//2+n//4 # Chai Wah Wu, Mar 05 2021

a(n) = 9*n - 2*a(n-1) - 2*a(n-2) - 2*a(n-3) - a(n-4) - 17 for n >= 4.
a(n) = a(n-2) + a(n-4) - a(n-6).
a(n) = A006368(n+1) - 1.
G.f.: (x^4+2*x^3+3*x^2+x+2)*x/((x^2+1)*(x-1)^2*(x+1)^2). - Alois P. Heinz, Mar 01 2021
E.g.f.: (cos(x) + (6*x - 1)*cosh(x) + (2 + 3*x)*sinh(x))/4. - Stefano Spezia, Mar 02 2021
From Bruno Berselli, Mar 05 2021: (Start)
a(n) = (12*n + 4 - (3*n + 3 - (-1)^(n/2))*(1 + (-1)^n))/8. Therefore:
a(4*k)   = 3*k;
a(4*k+1) = 6*k + 2;
a(4*k+2) = 3*k + 1;
a(4*k+3) = 6*k + 5. (End)

A223084 Trajectory of 80 under the map n-> A006369(n).

Original entry on oeis.org

80, 107, 143, 191, 255, 170, 227, 303, 202, 269, 359, 479, 639, 426, 284, 379, 505, 673, 897, 598, 797, 1063, 1417, 1889, 2519, 3359, 4479, 2986, 3981, 2654, 3539, 4719, 3146, 4195, 5593, 7457, 9943, 13257, 8838, 5892, 3928, 5237, 6983, 9311, 12415, 16553, 22071

Offset: 1

N. J. A. Sloane, Mar 22 2013

nonn

It is conjectured that this trajectory does not close on itself.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. H. Conway, On unsettleable arithmetical problems, Amer. Math. Monthly, 120 (2013), 192-198.

Cf. A006369, A006368, A182205.

Trajectories under A006368 and A006369: A180853, A217218, A185590, A180864, A028393, A028394, A094328, A094329, A028396, A028395, A217729, A182205, A223083-A223088, A185589, A185590.

f:=proc(N) if N mod 3 = 0 then 2*(N/3); elif N mod 3 = 2 then 4*((N+1)/3)-1; else 4*((N+2)/3)-3; fi; end;
t1:=[80];
for n from 1 to 100 do t1:=[op(t1),f(t1[nops(t1)])]; od:
t1;

t = {80}; While[n = t[[-1]]; s = Switch[Mod[n, 3], 0, 2*n/3, 1, (4*n - 1)/3, 2, (4*n + 1)/3]; Length[t] < 100 && ! MemberQ[t, s], AppendTo[t, s]]; t (* T. D. Noe, Mar 22 2013 *)
SubstitutionSystem[{n_ :> Switch[Mod[n, 3], 0, 2n/3, 1, (4n - 1)/3, , (4n + 1)/3]}, {80}, 60] // Flatten (* _Jean-François Alcover, Mar 01 2019 *)

cp's OEIS Frontend

A223083 Trajectory of 64 under the map n-> A006369(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

A265667 Permutation of nonnegative integers: a(n) = n + floor(n/3)*(-1)^(n mod 3).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Mathematica

Sage

Formula

A131743 Period 4: repeat [0, 1, 0, 2].

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

Mathematica

PARI

PARI

Python

Formula

A168222 a(n) = A006369(A006369(n)).

Views

Author

Keywords

Comments

Links

Programs

Mathematica

Formula

A349378 a(n) = A349376(n) + A349377(n).

Views

Author

Keywords

Links

Crossrefs

Programs

PARI

Formula

A178414 Least odd number in the Collatz (3x+1) preimage of odd numbers not a multiple of 3.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Views

Author

Keywords

Comments

Links

Crossrefs

A330530 Lexicographically earliest sequence of distinct positive integers such that the product of two consecutive terms is always divisible by 4.

Views

Author

Keywords

Comments

Examples

Links