A033493 -id:A033493 - cp's OEIS frontend

A275531 Values taken by the sum of numbers in Collatz trajectory (A033493), sorted into ascending order.

1, 3, 7, 15, 31, 36, 46, 49, 55, 63, 66, 67, 91, 106, 119, 127, 139, 145, 148, 186, 190, 197, 214, 235, 248, 255, 259, 274, 281, 288, 301, 302, 316, 325, 330, 339, 346, 357, 386, 393, 399, 413, 427, 442, 442, 452, 465, 497, 498, 500, 505, 509, 511, 519, 535, 540

Offset: 1

Jaroslav Krizek, Jul 31 2016

nonn

Values a(n) such that a(n) = a(n+1): 442, 609, 633, 724, 904, 925, ...

Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture

Cf. A033493, A275532.

Sort([&+[k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]: n in [1..2^10] | &+[k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]] le 2^10])

A275532 Possible values for sum of numbers in Collatz trajectory (A033493), sorted into ascending order.

Original entry on oeis.org

1, 3, 7, 15, 31, 36, 46, 49, 55, 63, 66, 67, 91, 106, 119, 127, 139, 145, 148, 186, 190, 197, 214, 235, 248, 255, 259, 274, 281, 288, 301, 302, 316, 325, 330, 339, 346, 357, 386, 393, 399, 413, 427, 442, 452, 465, 497, 498, 500, 505, 509, 511, 519, 535, 540, 557

Offset: 1

Jaroslav Krizek, Jul 31 2016

nonn

Values a(n) such that a(n) = A033493(x) = A033493(y) for distinct numbers x and y: 442, 609, 633, 724, 904, 925, ...

Cf. A033493, A275531.

Set(Sort([&+[k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]: n in [1..2^10] | &+[k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]] le 2^10]))

Take[#, 56] &@ Union@ Table[Total@ FixedPointList[Which[# == 1, 1, OddQ@ #, 3 # + 1, True, #/2] &, n] - 1, {n, 10^3}] (* Michael De Vlieger, Aug 02 2016 *)

A247717 Numbers n such that A033493(n)/A008908(n) is an integer.

Original entry on oeis.org

1, 5, 18, 58, 99, 153, 176, 228, 238, 240, 282, 341, 345, 421, 475, 585, 629, 712, 739, 779, 802, 815, 974, 995, 1078, 1088, 1237, 1257, 1262, 1290, 1346, 1398, 1424, 1459, 1673, 1694, 1724, 1731, 1802, 1811, 1916, 1928, 1988, 2170, 2222, 2260, 2272, 2275, 2317, 2365, 2397, 2410

Offset: 1

Derek Orr, Sep 22 2014

nonn

Equivalently, these are also numbers n such that the average of the numbers in the Collatz (3x+1) iteration of n is an integer.

Cf. A008908, A033493.

Tavg(n)=c=0;s=n;while(n!=1,if(n==Mod(0,2),n=n/2;c++;s+=n);if(n==Mod(1,2)&&n!=1,n=3*n+1;s+=n;c++));s/(c+1)
n=1;while(n<10^4,if(floor(Tavg(n))==Tavg(n),print1(n,", "));n++)

A254783 Numbers n such that A033493(n)/n is an integer.

Original entry on oeis.org

1, 57, 847, 1694, 3039, 3388, 3479, 6078, 6776, 6958, 13916, 27832, 55664, 111328, 236107, 246721, 311257, 493442, 622514, 986884, 1245028, 1328233, 1973768, 2052521, 2490056, 2656466, 3947536, 4105042, 4980112, 8210084

Offset: 1

Derek Orr, Feb 07 2015

If A033493(n)/n = M is even, then 2*n is a member of the sequence and A033493(2*n)/(2*n) = M/2 + 1.
a(31) > 10^7. - Derek Orr, Mar 12 2015
Sum of reciprocals seems to converge quickly to 1.0208... - Derek Orr, Mar 12 2015

Eric Weisstein's World of Mathematics, CollatzProblem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A033493.

a033493[n_] := Block[{f}, f[1] = 1; f[x_Integer?OddQ] := 3 x + 1; f[x_Integer?EvenQ] := x/2; -1 + Plus @@ FixedPointList[f, n]]; Select[Range[10^5], IntegerQ[a033493[#]/#] &] (* Michael De Vlieger, Feb 09 2015, after Alonso del Arte at A033493 *)

Tsum(n)=s=n;while(n!=1,if(n==Mod(0,2),n=n/2;s+=n);if(n==Mod(1,2)&&n!=1,n=3*n+1;s+=n));s
for(n=1,10^6,if(type(Tsum(n)/n)=="t_INT",print1(n,", ")))

A275114 Primes p for which the sum of the numbers in the Collatz iteration (A033493) of p is a prime.

Original entry on oeis.org

2, 67, 149, 163, 229, 359, 373, 401, 571, 719, 727, 827, 919, 941, 1031, 1049, 1129, 1153, 1201, 1283, 1307, 1319, 1433, 1453, 1627, 1637, 1987, 2017, 2089, 2137, 2237, 2267, 2281, 2351, 2543, 2617, 2731, 2819, 2851, 2861, 2927, 2969, 3191, 3253, 3581, 3671, 3719

Offset: 1

Jaroslav Krizek, Jul 17 2016

nonn

Primes p such that A033493(p) is a prime.

Prime terms from A225748.

			Prime 67 with Collatz trajectory (67, 202, 101, 304, 152, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1) is a term because A033493(67) = 1459 (prime).

Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture

Cf. A033493, A225748.

[n: n in [1..4000] | IsPrime(&+[k eq 1 select n else IsOdd(Self(k-1)) and not IsOne(Self(k-1)) select 3*Self(k-1)+1 else Self(k-1) div 2: k in [1..5*n]]) and IsPrime(n)];

Select[Prime@ Range@ 540, PrimeQ[Total@ FixedPointList[Which[# == 1, 1, EvenQ@ #, #/2, True, 3 # + 1] &, #] - 1] &] (* Michael De Vlieger, Jul 17 2016, after Alonso del Arte at A033493 *)

A070165 Irregular triangle read by rows giving trajectory of n in Collatz problem.

Original entry on oeis.org

1, 2, 1, 3, 10, 5, 16, 8, 4, 2, 1, 4, 2, 1, 5, 16, 8, 4, 2, 1, 6, 3, 10, 5, 16, 8, 4, 2, 1, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 8, 4, 2, 1, 9, 28, 14, 7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1, 10, 5, 16, 8, 4, 2, 1, 11, 34, 17, 52, 26, 13

Offset: 1

Eric W. Weisstein, Apr 23 2002

n-th row has A008908(n) entries (unless some n never reaches 1, in which case the triangle ends with an infinite row). [Escape clause added by N. J. A. Sloane, Jun 06 2017]
A216059(n) is the smallest number not occurring in n-th row; see also A216022.
Comment on the mp3 file from Gordon Charlton (the recording artist Beat Frequency). The piece uses the first 3242 terms (i.e. the first 100 hailstone sequences), with pitch modulus 36, duration modulus 2. Its musicality stems from the many repetitions and symmetries within the sequence, and in particular the infrequency of multiples of 3. This means that when the pitch modulus is a multiple of 12 the notes are predominantly in the symmetric octatonic scale, known to modern classical composers as the second of Messiaen's modes of limited transposition, and to jazz musicians as half-whole diminished. - N. J. A. Sloane, Jan 30 2019

			The irregular array a(n,k) starts:
n\k   0  1  2  3  4   5  6   7  8  9 10 11 12 13 14 15 16 17 18 19
1:    1
2:    2  1
3:    3 10  5 16  8   4  2   1
4:    4  2  1
5:    5 16  8  4  2   1
6:    6  3 10  5 16   8  4   2  1
7:    7 22 11 34 17  52 26  13 40 20 10  5 16  8  4  2  1
8:    8  4  2  1
9:    9 28 14  7 22  11 34  17 52 26 13 40 20 10  5 16  8  4  2  1
10:  10  5 16  8  4   2  1
11:  11 34 17 52 26  13 40  20 10  5 16  8  4  2  1
12:  12  6  3 10  5  16  8   4  2  1
13:  13 40 20 10  5  16  8   4  2  1
14:  14  7 22 11 34  17 52  26 13 40 20 10  5 16  8  4  2  1
15:  15 46 23 70 35 106 53 160 80 40 20 10  5 16  8  4  2  1
... Reformatted and extended by _Wolfdieter Lang_, Mar 20 2014

T. D. Noe, Rows n = 1..100 of triangle, flattened
Gordon Charlton ("Beat Frequency"), Hailstone Trajectory (mp3 file)
David Eisenbud and Brady Haran, UNCRACKABLE? The Collatz Conjecture, Numberphile Video, 2016.
David Rabahy, Hailstone Sequence presented as a spreadsheet
Anatoly E. Voevudko, File of first 10K Collatz sequences
Eric Weisstein's World of Mathematics, Collatz Problem
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370 (step), A008908 (row lengths), A033493 (row sums).
Cf. A220237 (sorted rows), A347270 (array), A192719.
Cf. A070168 (Terras triangle), A256598 (reduced triangle).
Cf. A254311, A257480 (and crossrefs therein).
Cf. A280408 (primes).

a070165 n k = a070165_tabf !! (n-1) !! (k-1)
a070165_tabf = map a070165_row [1..]
a070165_row n = (takeWhile (/= 1) $ iterate a006370 n) ++ [1]
a070165_list = concat a070165_tabf
-- Reinhard Zumkeller, Oct 07 2011

T:= proc(n) option remember; `if`(n=1, 1,
      [n, T(`if`(n::even, n/2, 3*n+1))][])
    end:
seq(T(n), n=1..15);  # Alois P. Heinz, Jan 29 2021

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Flatten[Table[Collatz[n], {n, 10}]] (* T. D. Noe, Dec 03 2012 *)

row(n, lim=0)={if (n==1, return([1])); my(c=n, e=0, L=List(n)); if(lim==0, e=1; lim=n*10^6); for(i=1, lim, if(c%2==0, c=c/2, c=3*c+1); listput(L, c); if(e&&c==1, break)); return(Vec(L)); } \\ Anatoly E. Voevudko, Mar 26 2016; edited by Michel Marcus, Aug 10 2021

def a(n):
    if n==1: return [1]
    l=[n, ]
    while True:
        if n%2==0: n/=2
        else: n = 3*n + 1
        if n not in l:
            l+=[n, ]
            if n<2: break
        else: break
    return l
for n in range(1, 101): print(a(n)) # Indranil Ghosh, Apr 14 2017

T(n,k) = T^{(k)}(n) with the k-th iterate of the Collatz map T with T(n) = 3*n+1 if n is odd and T(n) = n/2 if n is even, n >= 1. T^{(0)}(n) = n. k = 0, 1, ..., A008908(n) - 1. - Wolfdieter Lang, Mar 20 2014

Name specified and row length A-number corrected by Wolfdieter Lang, Mar 20 2014

A213909 Sum of all even numbers in Collatz (3x+1) trajectory of n.

Original entry on oeis.org

0, 2, 40, 6, 30, 46, 234, 14, 276, 40, 212, 58, 100, 248, 562, 30, 178, 294, 424, 60, 126, 234, 516, 82, 538, 126, 81178, 276, 366, 592, 80910, 62, 688, 212, 446, 330, 444, 462, 1894, 100, 81096, 168, 1090, 278, 416, 562, 80816, 130, 666, 588, 926, 178, 340

Offset: 1

Jayanta Basu, Mar 05 2013

nonn

			a(5)=30 since Collatz trajectory of 5 is 5,16,8,4,2,1.

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

Cf. A033493.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Total[Select[Collatz[n], EvenQ]], {n, 100}] (* T. D. Noe, Mar 05 2013 *)

A220237 Triangle read by rows: sorted terms of Collatz trajectories.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 5, 8, 10, 16, 1, 2, 4, 1, 2, 4, 5, 8, 16, 1, 2, 3, 4, 5, 6, 8, 10, 16, 1, 2, 4, 5, 7, 8, 10, 11, 13, 16, 17, 20, 22, 26, 34, 40, 52, 1, 2, 4, 8, 1, 2, 4, 5, 7, 8, 9, 10, 11, 13, 14, 16, 17, 20, 22, 26, 28, 34, 40, 52, 1, 2, 4, 5, 8, 10, 16

Offset: 1

Reinhard Zumkeller, Jan 03 2013

n-th row = sorted list of {A070165(n,k): k = 1..A006577(n)};

T(n,1) = 1 if Collatz conjecture is true.

			The table begins:
.   1:  [1]
.   2:  [1,2]
.   3:  [1,2,3,4,5,8,10,16]
.   4:  [1,2,4]
.   5:  [1,2,4,5,8,16]
.   6:  [1,2,3,4,5,6,8,10,16]
.   7:  [1,2,4,5,7,8,10,11,13,16,17,20,22,26,34,40,52]
.   8:  [1,2,4,8]
.   9:  [1,2,4,5,7,8,9,10,11,13,14,16,17,20,22,26,28,34,40,52]
.  10:  [1,2,4,5,8,10,16]
.  11:  [1,2,4,5,8,10,11,13,16,17,20,26,34,40,52]
.  12:  [1,2,3,4,5,6,8,10,12,16] .

Reinhard Zumkeller, Rows n = 1..120 of triangle, flattened
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006577 (row lengths), A025586(right edge), A033493 (row sums).

import Data.List (sort)
a220237 n k = a220237_tabf !! (n-1) !! (k-1)
a220237_row n = a220237_tabf !! (n-1)
a220237_tabf = map sort a070165_tabf

T:= proc(n) option remember; `if`(n=1, 1,
      sort([n, T(`if`(n::even, n/2, 3*n+1))])[])
    end:
seq(T(n), n=1..10);  # Alois P. Heinz, Oct 16 2021

Flatten[Table[Sort[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]],{n,12}]] (* Harvey P. Dale, Jan 28 2013 *)

A213916 Sum of all odd numbers in Collatz (3x+1) trajectory of n.

Original entry on oeis.org

1, 1, 9, 1, 6, 9, 54, 1, 63, 6, 47, 9, 19, 54, 132, 1, 36, 63, 95, 6, 22, 47, 117, 9, 120, 19, 20262, 54, 76, 132, 20194, 1, 153, 36, 94, 63, 91, 95, 450, 6, 20235, 22, 248, 47, 81, 117, 20163, 9, 140, 120, 204, 19, 59, 20262, 20426, 54, 305, 76, 411, 132, 178

Offset: 1

Jayanta Basu, Mar 05 2013

nonn

			a(5)=6 since Collatz trajectory of 5 is 5,16,8,4,2,1.

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

Cf. A033493, A213909.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[Total[Select[Collatz[n], OddQ]], {n, 100}] (* T. D. Noe, Mar 05 2013 *)

a(n) = A033493(n) - A213909(n).

A285098 Row sums of irregular triangle A070168.

Original entry on oeis.org

1, 3, 23, 7, 20, 29, 124, 15, 147, 30, 117, 41, 63, 138, 296, 31, 106, 165, 231, 50, 84, 139, 281, 65, 294, 89, 40616, 166, 212, 326, 40486, 63, 377, 140, 258, 201, 259, 269, 986, 90, 40589, 126, 588, 183, 253, 327, 40455, 113, 382, 344, 514, 141, 223, 40670, 41000, 222

Offset: 1

Indranil Ghosh, Apr 17 2017

nonn

a(n) is the sum of numbers in trajectory of Terras-modified Collatz problem with first number n.

			The 5th row of irregular triangle A070168 is [5, 8, 4, 2, 1] whose sum is 20. a(5) = 5 + 8 + 4 + 2 + 1 = 20.

Indranil Ghosh, Table of n, a(n) for n = 1..10000

Cf. A033493, A070168.

a[n_]:=If[n<2, 1, If[OddQ[n], (3n + 1)/2, n/2]]; Table[-1 + Plus @@ FixedPointList[a, n], {n, 100}]

def a(n):
    if n==1: return 1
    l=[n]
    while True:
        if n%2==0: n//=2
        else: n = (3*n + 1)//2
        l.append(n)
        if n<2: break
    return sum(l)
print([a(n) for n in range(1, 101)])

cp's OEIS Frontend

A275531 Values taken by the sum of numbers in Collatz trajectory (A033493), sorted into ascending order.

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A275532 Possible values for sum of numbers in Collatz trajectory (A033493), sorted into ascending order.

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Mathematica

A247717 Numbers n such that A033493(n)/A008908(n) is an integer.

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A254783 Numbers n such that A033493(n)/n is an integer.

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A275114 Primes p for which the sum of the numbers in the Collatz iteration (A033493) of p is a prime.

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Magma

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A070165 Irregular triangle read by rows giving trajectory of n in Collatz problem.

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A213909 Sum of all even numbers in Collatz (3x+1) trajectory of n.

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Mathematica

A220237 Triangle read by rows: sorted terms of Collatz trajectories.

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