A061641 -id:A061641 - cp's OEIS frontend

A187831 Smallest number m > n such that n occurs in Collatz trajectory starting with m; a(0) = 1 by convention.

1, 2, 3, 6, 5, 6, 12, 9, 9, 18, 11, 14, 24, 14, 18, 30, 17, 18, 36, 25, 22, 42, 25, 27, 48, 33, 28, 54, 36, 33, 60, 41, 42, 66, 36, 41, 72, 43, 39, 78, 41, 54, 84, 57, 50, 90, 47, 54, 96, 57, 66, 102, 56, 54, 108, 73, 57, 114, 59, 78, 120, 62, 82, 126, 75

Offset: 0

Reinhard Zumkeller, Jan 04 2013

nonn

A070165(a(n),k) = n for some k with 1 <= k <= A006577(a(n)).

			n = 10: row 11 of A070165 = [11,34,17,52,26,13,40,20,10,5,16,8,4,2,1],
therefore A070165(11,9) = 10 and a(10) = 11;
n = 11: rows 12 and 13 of A070165 don't contain 11, but 14 does:
row 12: [12,6,3,10,5,16,8,4,2,1],
row 13: [13,40,20,10,5,16,8,4,2,1],
row 14: [14,7,22,11,34,17,52,26,13,40,20,10,5,16,8,4,2,1],
therefore A070165(14,4) = 11: a(11) = 14.

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A177729, A061641.

Cf. A225840.

import Data.List (find)
import Data.Maybe (fromJust)
a187831 0 = 1
a187831 n = head $ fromJust $
        find (n `elem`) $ drop (fromIntegral n) a070165_tabf

mcollQ[n_,k_]:=MemberQ[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&],k]==True; Prepend[Table[i=n+1; While[!mcollQ[i,n],i++]; i,{n,64}],1] (* Jayanta Basu, May 27 2013 *)
Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Join[{1}, Table[k = n + 1; While[! MemberQ[Collatz[k], n], k++]; k, {n, 100}]] (* T. D. Noe, May 28 2013 *)

A127929 a(n) = A127928(n) mod 18.

Original entry on oeis.org

3, 7, 1, 1, 7, 1, 7, 7, 1, 1, 7, 1, 1, 1, 7, 7, 1, 7, 1, 7, 7, 7, 7, 1, 1, 7, 7, 7, 1, 1, 1, 7, 7, 1, 7, 1, 7, 7, 7, 7, 1, 1, 1, 7, 1, 7, 1, 7, 1, 7, 7, 7, 7, 1, 7, 1, 7, 7, 7, 1, 1, 7, 7, 1, 7, 7, 1, 7, 1, 1, 7, 1, 7, 1, 1, 7, 1, 7, 1, 7, 1, 7, 7, 7, 7, 7, 1

Offset: 1

Gary W. Adamson, Feb 07 2007

nonn

Aside from "3", all terms of A127928 must be 1 or 7 mod 18 (see A127928 for mod rules); but not all primes mod 1 or 7 are pure hailstone numbers. For example, the prime 61 == 7 mod 18 but 61 is impure. Conjecture: for large n, the numbers of 1 and 7 mod 18 terms are approximately equal.

			a(5) = 7 since A127928(5) = 43 and 43 == 7 mod 18.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Douglas J. Shaw, The Pure Numbers Generated by the Collatz Sequence, The Fibonacci Quarterly, Vol. 44, Number 3, August 2006, p. 194.

Cf. A127928, A127930, A061641, A127633, A006577, A066903.

Pure hailstone (Collatz) numbers that are also prime (i.e. the set A127928), mod 18.

More terms from Amiram Eldar, Feb 28 2020

A134191 Impure numbers in the Collatz (3x+1) iteration.

Original entry on oeis.org

2, 4, 5, 8, 10, 11, 13, 14, 16, 17, 20, 22, 23, 26, 28, 29, 31, 32, 34, 35, 38, 40, 41, 44, 46, 47, 49, 50, 52, 53, 56, 58, 59, 61, 62, 64, 65, 67, 68, 70, 71, 74, 76, 77, 80, 82, 83, 85, 86, 88, 89, 91, 92, 94, 95, 98, 100, 101, 103, 104, 106, 107, 110, 112, 113, 116, 118

Offset: 1

T. D. Noe, Oct 12 2007

nonn

Let f(k) be the trajectory of the Collatz iteration of the number k. Then Shaw calls a number n impure if n is in f(k) for some k < n. Shaw has an algorithm for finding congruences that the impure numbers satisfy.

			The Collatz trajectory of 3 is (3,10,5,16,8,4,2,1), showing that the numbers 4,5,8,10,16 are impure.

T. D. Noe, Table of n, a(n) for n=1..10000
Douglas J. Shaw, The Pure Numbers Generated by the Collatz Sequence, The Fibonacci Quarterly, Vol. 44, Number 3, August 2006, pp. 194-201.

c[n_] := If[EvenQ[n], n/2, 3n + 1]; nn=1000; t=Table[0,{nn}]; Do[If[t[[n]]==0, m=n; While[m=c[m]; If[nn>=m>n && t[[m]]==0, t[[m]]=n]; m>nn || t[[m]]>0]], {n,nn}]; Flatten[Position[t,_?(#>0&)]]

Complement of A061641.

A187108 Odd numbers not in the trajectory of a smaller number under the Collatz (3x+1) iteration.

Original entry on oeis.org

1, 3, 7, 9, 15, 19, 21, 25, 27, 33, 37, 39, 43, 45, 51, 55, 57, 63, 69, 73, 75, 79, 81, 87, 93, 97, 99, 105, 109, 111, 115, 117, 123, 127, 129, 133, 135, 141, 145, 147, 151, 153, 159, 163, 165, 169, 171, 177, 181, 183, 187, 189, 195, 199, 201, 207, 213, 217

Offset: 1

Jimin Park, Mar 05 2011

nonn

These are the odd numbers in A061641, which gives a more detailed explanation. - T. D. Noe, Mar 05 2011

Cf. A061641.

// AS3
var a:Array=new Array();
var i:int;
var n:int=0;
var ni:int;
var s:String='';
for (i=0;i<50;i++){
while(a[n]!=undefined) n++;
s+=String(2*n+1)+',';
a[n]=1;
ni=2*n+1;
while(ni>=2*n+1&&ni>1){
ni=3*ni+1;
while(ni%2==0)ni/=2;
a[(ni-1)/2]=1;
}
}
trace(s);

A235452 Take the union of all the sequences Collatz(i) for i <= n. The number a(n) is the largest of consecutive numbers beginning with 1.

Original entry on oeis.org

1, 2, 5, 5, 5, 6, 8, 8, 11, 11, 11, 14, 14, 14, 17, 17, 17, 18, 20, 20, 23, 23, 23, 24, 26, 26, 29, 29, 29, 32, 32, 32, 35, 35, 35, 36, 38, 38, 41, 41, 41, 42, 44, 44, 47, 47, 47, 50, 50, 50, 53, 53, 53, 54, 56, 56, 59, 59, 59, 62, 62, 62, 65, 65, 65

Offset: 1

Martin Y. Champel, Jan 10 2014

nonn

The Collatz sequence is also called the 3x+1 sequence.

			Let the C(n) function compute the Collatz sequence starting at n.
For n = 1, C(1) = {1} then term 1 is 1.
For n = 2, C(2) = {1,2} then term 2 is 2.
For n = 3, C(3) = {3,10,5,16,8,4,2,1} = {1,2,3,4,5,8,10,16} then it contains {1,2,3,4,5} but not {1,2,3,4,5,6} then term 3 is 5.
For n = 4, C(4) = C(3) then term 4 is 5.
For n = 5, C(5) = C(4) = C(3) then term 5 is 5.
For n = 6, C(6) = {1,2,3,4,5,6,8,10,16} then term 6 is 6.

Martin Y. Champel, Table of n, a(n) for n = 1..1000

Cf. A061641, A177729.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; countConsec[lst_] := Module[{cnt = 0, i = 1}, While[i <= Length[lst] && lst[[i]] == i, cnt++; i++]; cnt]; mx = 0; u = {}; Table[c = Collatz[n]; u = Union[u, c]; mx = Max[mx, countConsec[u]], {n, 65}] (* T. D. Noe, Feb 23 2014 *)

def A235452(n=100):
    a = set([])
    A235452 = {1: 1}
    for i in range(2, n):
        c = i
        a.add(c)
        while c != 1:
            if c % 2 == 1:
                c = 3 * c + 1
                a.add(c)
            c = c / 2
            a.add(c)
        k = 1
        while k in a:
            k += 1
        A235452[i] = k - 1
    return A235452
seq_map = A235452()
for n in range(1, len(seq_map) + 1):
    print(seq_map[n], end=", ")

A336256 The cardinalities of the sets A(n), where A(0) is the empty set and A(n+1) is the union of A(n) and the Collatz orbit of the smallest natural number missing in A(n).

Original entry on oeis.org

0, 1, 4, 9, 10, 20, 23, 24, 33, 34, 39, 42, 43, 46, 141, 142, 145, 146, 149, 161, 162, 170, 173, 174, 179, 180, 187, 190, 191, 204, 205, 208, 209, 212, 220, 221, 230, 231, 232, 239, 240, 243, 244, 247, 252, 253, 256, 257, 260, 261, 262, 267, 270, 271, 284, 285

Offset: 0

Markus Sigg, Aug 08 2020

nonn

Cf. A061641, A336938.

firstMiss(A) = { my(i); if(#A == 0 || A[1] > 0, return(0)); for(i = 1, A[#A] + 1, if(!setsearch(A,i), return(i))); };
iter(A) = { my(a = firstMiss(A)); while(!setsearch(A,a), A = setunion(A, Set([a])); a = if(a % 2, 3*a+1, a/2)); A; };
makeVec(m) = { my(v = [], A = Set([]), i); for(i = 1, m, v = concat(v, #A); if (i < m, A = iter(A))); v; };
makeVec(56)

A336992 The number of gaps in the sets A(n), where A(0) is the empty set and A(n+1) is the union of A(n) and the Collatz orbit of the smallest natural number missing in A(n).

Original entry on oeis.org

0, 0, 1, 3, 3, 9, 9, 8, 12, 12, 14, 15, 15, 16, 100, 99, 100, 100, 101, 108, 108, 112, 111, 110, 110, 110, 110, 111, 110, 116, 115, 115, 115, 116, 120, 120, 124, 123, 122, 122, 121, 122, 122, 123, 125, 124, 125, 125, 126, 125, 125, 127, 127, 126, 133, 133

Offset: 0

Markus Sigg, Aug 10 2020

nonn

The number of gaps would be relevant for sparse representations of the sets A(n), which may be of use for a numerical verification of the Collatz conjecture up to a given number.

Markus Sigg, Table of n, a(n) for n = 0..999

Cf. A061641, A336938.

firstMiss(A) = { my(i); if(#A == 0 || A[1] > 0, return(0)); for(i = 1, A[#A] + 1, if(!setsearch(A, i), return(i))); };
iter(A) = { my(a = firstMiss(A)); while(!setsearch(A, a), A = setunion(A, Set([a])); a = if(a % 2, 3*a+1, a/2)); A; };
nGaps(A) = { my(i,c=0); for (i=2, #A, if (A[i-1] < A[i]-1, c = c+1;)); c; };
makeVec(m) = { my(v = [], A = Set([]), i); for(i = 1, m, v = concat(v, nGaps(A)); if (i < m, A = iter(A))); v; };
makeVec(57)

A375852 Numbers congruent to {0, 1, 3, 6, 7, 9, 12, 15} mod 18.

Original entry on oeis.org

0, 1, 3, 6, 7, 9, 12, 15, 18, 19, 21, 24, 25, 27, 30, 33, 36, 37, 39, 42, 43, 45, 48, 51, 54, 55, 57, 60, 61, 63, 66, 69, 72, 73, 75, 78, 79, 81, 84, 87, 90, 91, 93, 96, 97, 99, 102, 105, 108, 109, 111, 114, 115, 117, 120, 123, 126, 127, 129, 132, 133, 135, 138, 141, 144, 145, 147, 150

Offset: 1

Jules Beauchamp, Aug 31 2024

Appears to be the union of A061641 (pure numbers in the Collatz (3x+1) iteration, also called pure hailstone numbers) and A309180 (unsuspected numbers to check in the Collatz conjecture).

The differences are periodic: 1, 2, 3, 1, 2, 3, 3, 3.

Index entries for linear recurrences with constant coefficients, signature (2,-2,2,-2,2,-2,2,-1).

Cf. A061641, A309180.

Select[Range[0, 200], MemberQ[{0, 1, 3, 6, 7, 9, 12, 15}, Mod[#, 18]] &] (* Amiram Eldar, Aug 31 2024 *)

From Stefano Spezia, Sep 03 2024: (Start)
G.f.: x^2*(1 + x + 2*x^2 - x^3 + 3*x^4 + 3*x^6)/((1 - x)^2*(1 + x^2 + x^4 + x^6)).
E.g.f.: ((9*x - 14)*cosh(x) + sin(x) + 2*sqrt(2)*cosh(x/sqrt(2))*sin(x/sqrt(2)) + (9*x - 14)*sinh(x) + 2*(6 + cos(x) + (sqrt(2)*cos(x/sqrt(2)) + sin(x/sqrt(2)))*sinh(x/sqrt(2))))/4. (End)

cp's OEIS Frontend

A187831 Smallest number m > n such that n occurs in Collatz trajectory starting with m; a(0) = 1 by convention.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

A127929 a(n) = A127928(n) mod 18.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

Extensions

A134191 Impure numbers in the Collatz (3x+1) iteration.

Views

Author

Keywords

Comments

Examples

Links

Programs

Mathematica

Formula

A187108 Odd numbers not in the trajectory of a smaller number under the Collatz (3x+1) iteration.

Views

Author

Keywords

Comments

Crossrefs

Programs

ActionScript

A235452 Take the union of all the sequences Collatz(i) for i <= n. The number a(n) is the largest of consecutive numbers beginning with 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Python

A336256 The cardinalities of the sets A(n), where A(0) is the empty set and A(n+1) is the union of A(n) and the Collatz orbit of the smallest natural number missing in A(n).

Views

Author

Keywords

Crossrefs

Programs

PARI

A336992 The number of gaps in the sets A(n), where A(0) is the empty set and A(n+1) is the union of A(n) and the Collatz orbit of the smallest natural number missing in A(n).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

PARI

A375852 Numbers congruent to {0, 1, 3, 6, 7, 9, 12, 15} mod 18.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula