A070165 -id:A070165 - cp's OEIS frontend

A111329 Number of partitions of T where T = (3n + 1) if n is even and T=(3n + 1)/2 if n is odd.

2, 15, 7, 101, 22, 490, 56, 1958, 135, 6842, 297, 21637, 627, 63261, 1255, 173525, 2436, 451276, 4565, 1121505, 8349, 2679689, 14883, 6185689, 26015, 13848650, 44583, 30167357, 75175, 64112359, 124754, 133230930, 204226, 271248950, 329931

Offset: 1

Parthasarathy Nambi, Nov 04 2005

nonn

			If n=1 then T = 2 and a(1) = 2.

Jeffrey C. Lagarias The 3x+1 problem: An annotated bibliography arXiv:math/0309224 [math.NT], 2003-2011.
Jeffrey C. Lagarias, "The Problem and Its Generalizations." Amer. Math. Monthly 92, 3-23, 1985.
Eric Weisstein's World of Mathematics, Collatz Problem

Cf. A000546, A070165, A006577.

f[n_] := If[EvenQ[n], PartitionsP[3n + 1], PartitionsP[(3n + 1)/2]]; Table[ f[n], {n, 35}] (* Robert G. Wilson v, Nov 07 2005 *)

a(n) = A000041(A165355(n-1)). [Reinhard Zumkeller, Nov 19 2009]

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

Original entry on oeis.org

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 *)

A207675 Numbers such that not all divisors occur in their Collatz trajectories.

Original entry on oeis.org

9, 15, 18, 21, 27, 30, 33, 35, 36, 39, 42, 45, 51, 54, 55, 57, 60, 63, 66, 69, 70, 72, 75, 77, 78, 81, 84, 85, 87, 90, 91, 93, 95, 96, 99, 102, 105, 108, 110, 111, 114, 115, 117, 119, 120, 121, 123, 125, 126, 129, 132, 133, 135, 138, 140, 141, 143, 144, 145

Offset: 1

Reinhard Zumkeller, Feb 20 2012

nonn

			3 is a divisor of 9, not occurring in A033479 - therefore 9 is a term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A027750, A070165, A006370, A207674 (complement).

import Data.List (intersect)
a207675 n = a207675_list !! (n-1)
a207675_list = filter
   (\x -> a027750_row x `intersect` a070165_row x /= a027750_row x) [1..]

coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]; Select[Range[145],Complement[Divisors[#],coll[#]]!={}&] (* Jayanta Basu, May 27 2013 *)

A220139 The highest value of the Collatz iteration (3x+1) starting at a(n-1) + 1, with a(1) = 1.

Original entry on oeis.org

1, 2, 16, 52, 160, 9232, 27700, 83104, 599056, 1797172, 5391520, 38862808, 131161984, 393485956, 1180457872, 3541373620, 10624120864, 87327950740, 261983852224, 785951556676, 2357854670032, 7553654536192, 22660963608580, 67982890825744, 203948672477236

Offset: 1

T. D. Noe, Jan 02 2013

nonn

The length of the trajectory of a(n) is A220140(n).

			The Collatz trajectory of 2 + 1 is (3, 10, 5, 16, 8, 4, 2, 1). Hence, a(3) = 16. The trajectory of 16 + 1 is (17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1). Hence, a(4) = 52.

T. D. Noe, Table of n, a(n) for n = 1..500

Cf. A070165 (trajectory of n).

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; c = {1}; t = {}; Do[AppendTo[t, Max[c]]; c = Collatz[t[[-1]] + 1], {30}]; t

A225843 Integral parts of sums of the reciprocals of the Collatz (3x+1) sequence starting with n.

Original entry on oeis.org

1, 1, 2, 1, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 1, 3, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3

Offset: 1

Reinhard Zumkeller, May 16 2013

nonn

a(n) = floor(sum(1/A070165(n,k): k = 1..A006577(n)));
conjecture: a(n) <= 3;
a(n) = 1 iff n = 2^k: a(A000079(n)) = 1, a(A057716(n)) > 1.

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

a225843 = floor . sum . map (recip . fromIntegral) . a070165_row

a(n) = floor(A225761(n)/A225784(n)).

A238192 In the Collatz (3x+1) iteration of n, the last odd number before 1, or 0 if there is no such number.

Original entry on oeis.org

0, 0, 5, 0, 5, 5, 5, 0, 5, 5, 5, 5, 5, 5, 5, 0, 5, 5, 5, 5, 21, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 5, 5, 5, 5, 5, 5, 5, 5, 5, 21, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 0, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 85, 5, 5, 5, 5, 5, 5, 5, 5, 21, 85, 5

Offset: 1

T. D. Noe, Feb 21 2014

nonn

Another version of A237660. The only terms appearing here are 0, 5, 21, 85, ..., which is A002450 without 1.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A002450 ((4^n-1)/3), A070165 (Collatz trajectories).

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[c = Collatz[n]; co = Select[c, OddQ]; If[Length[co] == 1, 0, co[[-2]]], {n, 100}]

A254068 Irregular triangle T read by rows in which the entry in row n and column k is given by T(n,k) = 4*A253676(n,k) - 3, k = 1..A253720(n), assuming the 3x+1 (or Collatz) conjecture.

Original entry on oeis.org

1, 5, 1, 9, 17, 13, 5, 1, 13, 5, 1, 17, 13, 5, 1, 21, 1, 25, 29, 17, 13, 5, 1, 29, 17, 13, 5, 1, 33, 25, 29, 17, 13, 5, 1, 37, 17, 13, 5, 1, 41, 161, 121, 137, 233, 593, 445, 377, 425, 2429, 3077, 577, 433, 325, 61, 53, 5, 1, 45, 17, 13, 5, 1

Offset: 1

L. Edson Jeffery, May 03 2015

Definitions: Let v(y) denote the 2-adic valuation of y. Let N_1 denote the set of odd natural numbers. Let F : N_1 -> N_1 be the map defined by F(x) = (3*x + 1)/2^v(3*x + 1) (cf. A075677). Let F^(k)(x) denote k-fold iteration of F and defined by the recurrence F^(k)(x) = F(F^(k-1)(x)), k>0, with initial condition F^(0)(x) = x.
This triangle can be constructed by restricting the initial values to the numbers 4*n - 3, iterating F until 1 is reached (assuming the 3x+1 conjecture) and removing all iterates not congruent to 1 modulo 4. Equivalently, for each n, this is accomplished by iterating (until 1 is reached, assuming the 3x+1 conjecture) the function S defined in A257480 to get the triangle A253676, and finally taking T(n,k) = 4*A253676(n,k) - 3.
Conjecture: For each natural number n, there exists a k >= 0, such that F^k(4*n - 3) = 1.
Theorem 1: Conjecture 1 is equivalent to the 3x+1 (or Collatz) conjecture.
Proof: See A257480.

			T begins:
   1
   5   1
   9  17  13   5   1
  13   5   1
  17  13   5   1
  21   1
  25  29  17  13   5   1
  29  17  13   5   1
  33  25  29  17  13   5   1
  37  17  13   5   1
  41 161 121 137 233 593 445 377 425 2429 3077 577 433 325 61 53 5 1
  45  17  13   5   1
  49  37  17  13   5   1
  53   5   1
  57  65  49  37  17  13   5   1

Cf. A014682, A070165, A075677, A256598, A257480, A253676, A254070.

v[x_] := IntegerExponent[x, 2]; f[x_] := (3*x + 1)/2^v[3*x + 1]; s[n_] := NestWhileList[(3 + (3/2)^v[1 + f[4*# - 3]]*(1 + f[4*# - 3]))/6 &, n, # > 1 &]; t = Table[4*s[n] - 3, {n, 1, 15}]; Flatten[t] (* Replace Flatten with Grid to display the triangle *)

A258822 Number of times that k iterations of n under the '3x+1' map yield k for some k.

Original entry on oeis.org

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

Offset: 1

Derek Orr, Jun 11 2015

nonn

This sequence uses the definition in A006370: if n is odd, n -> 3*n+1, if n is even, n -> n/2.

The number 3 appears first at a(63105). Do all nonnegative numbers appear? See A258824.

			For n = 6, the '3x+1' map is as follows: 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1. Here the number of iterations is 8. However, after the k-th iteration, the result does not equal k. Thus a(6) = 0.
For n = 7, the '3x+1' map is as follows: 7 -> 22 -> 11 -> 34 -> 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1. Only after 10 iterations do we arrive at 10. Since this is the only time this happens, a(7) = 1.

Cf. A006370, A258772, A070165.

A258822[n_]:=Count[MapIndexed[{#1}==#2-1&,NestWhileList[If[OddQ[#],3#+1,#/2]&,n,#>1&]],True];Array[A258822,100] (* Paolo Xausa, Nov 06 2023 *)

Tvect(n)=v=[n]; while(n!=1, if(n%2, k=(3*n+1); v=concat(v, k); n=k); if(!(n%2), k=n/2; v=concat(v, k); n=k)); v
for(n=1, 200, d=Tvect(n); c=0; for(i=1, #d, if(d[i]==i-1, c++)); print1(c, ", "))

A334040 Number of odd numbers larger than n in the Collatz trajectory of n.

Original entry on oeis.org

0, 0, 1, 0, 0, 0, 3, 0, 3, 0, 2, 0, 0, 1, 3, 0, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 38, 0, 0, 2, 36, 0, 0, 0, 1, 0, 0, 0, 4, 0, 35, 0, 2, 0, 0, 1, 34, 0, 0, 0, 1, 0, 0, 33, 35, 0, 1, 0, 3, 0, 0, 32, 33, 0, 0, 0, 1, 0, 0, 0, 31, 0, 33, 0, 2, 0, 0, 2, 4, 0, 0, 31, 32

Offset: 1

Hamid Kulosman, May 11 2020

nonn

			For n=7 the Collatz process is: 7,22,(11),34,(17),52,26,(13),40,20,10,5,16,8,4,2,1. The numbers in the parentheses are odd numbers in the Collatz process for n=7 that are bigger than 7. There are three of them, hence a(7)=3.

Markus Sigg, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harvey P. Dale).
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A006577, A070165, A078719, A286380

Table[Count[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&],?(#>n&&OddQ[#]&)],{n,90}] (* _Harvey P. Dale, Aug 03 2023 *)

f(n) = if (n%2, 3*n+1, n/2);
a(n) = {my(nb = 0, m=n); while ((n=f(n)) != 1, if ((n % 2) && (n>m), nb++)); nb;} \\ Michel Marcus, Jun 01 2020

A341231 Irregular triangle read by rows giving trajectory from n to reach 1 under the map A245471.

Original entry on oeis.org

1, 2, 1, 3, 4, 2, 1, 4, 2, 1, 5, 14, 7, 8, 4, 2, 1, 6, 3, 4, 2, 1, 7, 8, 4, 2, 1, 8, 4, 2, 1, 9, 26, 13, 22, 11, 28, 14, 7, 8, 4, 2, 1, 10, 5, 14, 7, 8, 4, 2, 1, 11, 28, 14, 7, 8, 4, 2, 1, 12, 6, 3, 4, 2, 1, 13, 22, 11, 28, 14, 7, 8, 4, 2, 1, 14, 7, 8, 4, 2, 1

Offset: 1

Rémy Sigrist, Feb 07 2021

A340873 gives row lengths.

A341235 gives greatest terms.

			Table begins:
    1;
    2, 1;
    3, 4, 2, 1;
    4, 2, 1;
    5, 14, 7, 8, 4, 2, 1;
    6, 3, 4, 2, 1;
    7, 8, 4, 2, 1;
    8, 4, 2, 1;
    9, 26, 13, 22, 11, 28, 14, 7, 8, 4, 2, 1;
    10, 5, 14, 7, 8, 4, 2, 1;
    11, 28, 14, 7, 8, 4, 2, 1;
    12, 6, 3, 4, 2, 1;
    13, 22, 11, 28, 14, 7, 8, 4, 2, 1;
    14, 7, 8, 4, 2, 1;
    15, 16, 8, 4, 2, 1;
    16, 8, 4, 2, 1;
    ...

Rémy Sigrist, Table of n, a(n) for n = 1..9098 (rows 1..500)

Cf. A070165, A245471, A340873, A341235.

row(n) = { my (r=[n]); while (n>1, r=concat(r, n=if (n%2, bitxor(n, 2*n+1), n/2))); r }

T(n, 1) = n.

T(n, A340873(n)) = 1.

cp's OEIS Frontend

A111329 Number of partitions of T where T = (3n + 1) if n is even and T=(3n + 1)/2 if n is odd.

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

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A207675 Numbers such that not all divisors occur in their Collatz trajectories.

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A220139 The highest value of the Collatz iteration (3x+1) starting at a(n-1) + 1, with a(1) = 1.

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A225843 Integral parts of sums of the reciprocals of the Collatz (3x+1) sequence starting with n.

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A238192 In the Collatz (3x+1) iteration of n, the last odd number before 1, or 0 if there is no such number.

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A254068 Irregular triangle T read by rows in which the entry in row n and column k is given by T(n,k) = 4*A253676(n,k) - 3, k = 1..A253720(n), assuming the 3x+1 (or Collatz) conjecture.

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A258822 Number of times that k iterations of n under the '3x+1' map yield k for some k.

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