A078719 -id:A078719 - cp's OEIS frontend

A087228 a(n) is the smallest number k such that the LCM of the terms of the Collatz trajectory of k has n distinct prime factors.

2, 5, 3, 17, 11, 7, 9, 33, 67, 57, 59, 39, 105, 185, 191, 123, 225, 219, 239, 159, 319, 283, 251, 167, 335, 111, 297, 175, 233, 155, 103, 91, 107, 71, 31, 41, 27, 193, 129, 231, 171, 463, 327, 411, 859, 731, 487, 649, 639, 1153, 1563, 1607, 1071, 1215, 1307, 871, 1161

Offset: 1

Labos Elemer, Aug 28 2003

nonn

			a(10)=57 because 57 is the smallest number such that the LCM of the terms in its Collatz trajectory has 10 different prime factors: A082226(57) = 864203580240 = 2^4*3*5*7*11*13*17*19*37*43.

Cf. A006370, A087226, A086227, A078719, A001221.

c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1)c[1]=1; fpl[x_] := Delete[FixedPointList[c, x], -1] ef[x_] := Length[FactorInteger[Apply[LCM, fpl[x]]]] t=Table[0, {256}]; Do[s=ef[n]; If[s<257&&t[[s]]==0, t[[s]]=n], {n, 1, 1000}]; t

a(n) = Min{k; A087227(k)=n}, where A087227(k) = A001221(A087226(k)); A087226(k) = lcm(terms in Collatz trajectory of k).

Edited by Jon E. Schoenfield, Jul 09 2018

A282408 Numbers k for which the number of odd members and the number of even members in the Collatz (3x+1) trajectory are both a perfect square.

Original entry on oeis.org

1, 2, 16, 17, 322, 323, 512, 1595, 1598, 1599, 1609, 1614, 1615, 1627, 1641, 1643, 1657, 1663, 1776, 1780, 1781, 1784, 1786, 2176, 2208, 2216, 2218, 2240, 2256, 2260, 2261, 2274, 2275, 2400, 2408, 2410, 2416, 2417, 2420, 2421, 3844, 3845, 3846, 3848, 3850, 3852

Offset: 1

Michel Lagneau, Feb 14 2017

nonn

Or numbers m such that A078719(m) and A006666(m) are both a perfect square.
For k < 5*10^6, the fourteen distinct pairs of squares in the order of appearance are: (1, 0), (1, 1), (1, 4), (4, 9), (36, 64), (1, 9), (25, 49), (4, 16), (16, 36), (9, 25), (1, 16), (81, 144), (4, 25) and (64, 121).
The numbers 2^(m^2) = A002416(m) are in the sequence, and the corresponding pairs of squares are (1, m^2).
Number of terms <= 10^h: 2, 4, 7, 202, 203, 474, 20888, etc. Robert G. Wilson v, Feb 14 2017

			17 is in the sequence because the Collatz trajectory is 17 -> 52 -> 26 -> 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 => the number of odd members is 4 = 2^2 and number of even members is 9 = 3^2.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A000079, A002416, A078719, A006666, A006667.

nn:=10^6:
for n from 1 to 10000 do:
  m:=n:i1:=1:i2:=0:
   for i from 1 to nn while(m<>1) do:
    if irem(m,2)=0
     then
     m:=m/2:i2:=i2+1:
     else
     m:=3*m+1:i1:=i1+1:
    fi:
   od:
    if sqrt(i1)=floor(sqrt(i1)) and sqrt(i2)=floor(sqrt(i2))
     then
     printf(`%d, `,n):
     else
    fi:
od:

fQ[n_] := Block[{m = n, e = 0, o = 1}, While[m > 1, If[OddQ@ m, m = 3 m + 1; o++, m /= 2; e++]]; IntegerQ@ Sqrt@ e && IntegerQ@ Sqrt@ o]; Select[ Range@ 3855, fQ] (* Robert G. Wilson v, Feb 14 2017 *)
memsqQ[n_]:=Module[{col=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#!=1&]}, AllTrue[ {Sqrt[ Count[ col, ?(EvenQ[#]&)]],Sqrt[Count[col,?(OddQ[ #]&)]]},IntegerQ]]; Select[Range[4000],memsqQ] (* The program uses the AllTrue function from Mathematica version 10 *) (* Harvey P. Dale, Sep 04 2018 *)

A282409 Numbers n for which the number of odd members and the number of even members in the Collatz (3x+1) trajectory are both prime.

Original entry on oeis.org

3, 9, 10, 12, 13, 22, 23, 40, 42, 73, 88, 90, 92, 93, 114, 115, 118, 119, 144, 148, 149, 152, 154, 162, 163, 164, 165, 166, 192, 208, 212, 213, 226, 227, 251, 295, 318, 319, 350, 351, 576, 592, 596, 597, 608, 616, 618, 625, 640, 642, 643, 648, 650, 652, 653

Offset: 1

Michel Lagneau, Feb 14 2017

nonn

Or numbers m such that A078719(m) and A006666(m) are both prime.

The distinct pairs of primes in the order of appearance are: (3, 5), (7, 13), (2, 5), (3, 7), (5, 11), (2, 7), (43, 73), (5, 13), (11, 23), (7, 17), (41, 71), (3, 11), (23, 43), (19, 37),...

			13 is in the sequence because its Collatz trajectory is 13 -> 40 -> 20 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 which contains 3 odd members and 7 even members.

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A078719, A006666, A006667.

nn:=10^6:
for n from 1 to 500 do:
  m:=n:i1:=1:i2:=0:
   for i from 1 to nn while(m<>1) do:
    if irem(m,2)=0
     then
     m:=m/2:i2:=i2+1:
     else
     m:=3*m+1:i1:=i1+1:
    fi:
   od:
     if isprime(i1) and isprime(i2)
     then
     printf(`%d, `,n):
     else
    fi:od:

is(n)=my(e,o=1); while(n>1, n=if(n%2, o++; 3*n+1, e++; n/2)); isprime(e) && isprime(o) \\ Charles R Greathouse IV, Feb 14 2017

from sympy import isprime
def a(n):
    l=[n]
    while True:
        if n%2==0: n//=2
        else: n = 3*n + 1
        l.append(n)
        if n<2: break
    o=list(filter(lambda i: i%2==1, l))
    e=list(filter(lambda i: i%2==0, l))
    return [o, e]
print([n for n in range(2, 1001) if isprime(len(a(n)[0])) and isprime(len(a(n)[1]))]) # Indranil Ghosh, Apr 14 2017

A316783 Seed values of hailstone sequences for which, by the time they first reach 1, the ratio of odd terms to even terms is exactly 1/2. (The seed value itself is included in the ratio calculation.)

Original entry on oeis.org

4, 5, 6, 11, 14, 15, 18, 19, 25, 33, 43, 57, 59, 78, 79, 105, 135, 139, 185, 187, 191, 246, 247, 249, 254, 255, 329, 338, 339, 359, 427, 438, 439, 443, 450, 451, 478, 479, 569, 585, 590, 591, 601, 636, 637, 638, 758, 759, 767, 779, 786, 787, 801, 849, 850, 851

Offset: 1

Matt Enlow, Jul 13 2018

nonn

1/2 is the most common odd/even ratio for these sequences. For seed values from 2 to 2^17, it is the odd/even ratio for 4454 of those sequences. The next most common ratio is 5/11, with 1834 sequences. (This is all empirical observation, using Mathematica.)

			The hailstone sequence starting with 11 (and ending at 1) is 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1. Five of those terms are odd, and the other ten are even, giving an odd/even ratio of 1/2. Therefore 11 is a term in this sequence.

Cf. A006370, A006666, A078719.

b:= proc(n) option remember; `if`(n=1, [1, 0],
      `if`(n::odd, [1, 0]+b(3*n+1), [0, 1]+b(n/2)))
    end:
a:= proc(n) option remember; local k; for k from a(n-1)
      +1 while (l-> 2*l[1]<>l[2])(b(k)) do od; k
    end: a(0):=0:
seq(a(n), n=1..100);  # Alois P. Heinz, Aug 02 2018

collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1]&, n, # != 1 &];
A316783 = Select[Range[2, 1000], With[{c = Mod[collatz[#], 2]}, 2 Total[c] == Total[1 - c]] &]

is(n) = my(x=n, even=0, odd=0); while(1, if(x%2==0, x=x/2; even++, x=3*x+1; odd++); if(x==1, odd++; break)); odd/even==1/2 \\ Felix Fröhlich, Jul 13 2018

{ k : A006666(k)/A078719(k) = 2 }. - Alois P. Heinz, Aug 02 2018

A330732 a(n) is the smallest number whose odd hailstone sequence has length n (including 1 and itself) and is in descending order.

Original entry on oeis.org

1, 5, 13, 17, 45, 241, 321, 1713, 9137, 24365, 36033, 173261, 231681, 630033, 1642769, 4380717, 11715629, 31241677, 41655569, 111081517, 148108689, 789913009, 1053217345, 1404289793, 3744772781, 9986060749, 13314747665, 35505993773, 94682650061, 128599099649

Offset: 1

Xiangyu Chen, Dec 28 2019

nonn

The hailstone (Collatz) sequence of a number is defined by repeated operations of f(x), where f(x)=x/2 if x is even, and f(x)=3x+1 if x is odd. The odd numbers of a number's hailstone sequence are also called its odd hailstone sequence.
The odd hailstone sequence of 45 [45, 17, 13, 5, 1] contains the first five terms of this sequence.
Note that a(n+1) isn't necessarily greater than a(n).

			Consider a(8): [1713, 1285, 241, 181, 17, 13, 5, 1]; the odd hailstone sequence of 1713 is descending and has length 8. 1713 is the lowest such number.

Kevin P. Thompson, Table of n, a(n) for n = 1..36
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A078719, A092893, A182078.

With[{s = Array[If[AllTrue[Differences@ #, # < 0 &], Length@ #, 0] &@ Select[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, #, # > 1 &], OddQ] &, 10^5]}, Array[FirstPosition[s, #][[1]] &, Max@ s]] (* Michael De Vlieger, Jan 01 2020 *)

is(k, n) = {my(x=k, v=List([])); while(x>1, if(x%2==0, x=x/2, listput(v, x); x=3*x+1)); 1+#v==n&&v==vecsort(v, , 4); }
a(n) = {k=1; while(is(k, n)==0, k+=2); k; } \\ Jinyuan Wang, Dec 31 2019

a(23)-a(30) from Kevin P. Thompson, Jul 23 2022

A375888 Rectangular array: row n shows all k such that n is the number of rises in the trajectory of k in the Collatz problem.

Original entry on oeis.org

1, 2, 5, 4, 10, 3, 8, 20, 6, 17, 16, 21, 12, 34, 11, 32, 40, 13, 35, 22, 7, 64, 42, 24, 68, 23, 14, 9, 128, 80, 26, 69, 44, 15, 18, 25, 256, 84, 48, 70, 45, 28, 19, 49, 33, 512, 85, 52, 75, 46, 29, 36, 50, 65, 43, 1024, 160, 53, 136, 88, 30, 37, 51, 66, 86, 57

Offset: 0

Clark Kimberling, Sep 11 2024

Assuming that the Collatz conjecture (also known as the 3x+1 conjecture) is true, this is a permutation of the positive integers; viz., every positive integer occurs exactly once. Conjecture: every row contains a pair of consecutive integers.

			Corner:
   1     2     4     8    16    32    64   128   256   512  1024
   5    10    20    21    40    42    80    84    85   160   168
   3     6    12    13    24    26    48    52    53    96   104
  17    34    35    68    69    70    75   136   138   140   141
  11    22    23    44    45    46    88    90    92    93   176
   7    14    15    28    29    30    56    58    60    61   112
   9    18    19    36    37    38    72    74    76    77    81
6 is in row 2 because the trajectory, (6, 3, 10, 5, 16, 4, 2, 1), has exactly 2 rises: 3 to 10, and 5 to 16.

Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A000027, A000079 (row 1), A092893 (column 1), A006667, A070265, A078719.

Cf. A354236.

t = Table[Count[Differences[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]], ? Positive], {n, 2048}]; (* after _Harvey P. Dale, A006667 *)
r[n_] := Flatten[Position[t, n - 1]];
Column[Table[r[n], {n, 1, 21}]] (* array *)
u = Table[r[k][[n + 1 - k]], {n, 1, 12}, {k, 1, n}]
Flatten[u] (* sequence *)

Transpose of the array in A354236.

cp's OEIS Frontend

A087228 a(n) is the smallest number k such that the LCM of the terms of the Collatz trajectory of k has n distinct prime factors.

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A282408 Numbers k for which the number of odd members and the number of even members in the Collatz (3x+1) trajectory are both a perfect square.

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A282409 Numbers n for which the number of odd members and the number of even members in the Collatz (3x+1) trajectory are both prime.

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A316783 Seed values of hailstone sequences for which, by the time they first reach 1, the ratio of odd terms to even terms is exactly 1/2. (The seed value itself is included in the ratio calculation.)

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A330732 a(n) is the smallest number whose odd hailstone sequence has length n (including 1 and itself) and is in descending order.

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Mathematica

PARI

Extensions

A375888 Rectangular array: row n shows all k such that n is the number of rises in the trajectory of k in the Collatz problem.

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Mathematica

Formula