A057688 -id:A057688 - cp's OEIS frontend

A057689 Maximal term in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if no such term exists.

16, 66, 50, 672, 20372, 494, 36918, 1404, 12210, 4248, 5070, 1682, 1850, 2210, 35882, 102720, 94484303672, 30084, 178992, 5330, 246560, 6890, 294253314, 8416400, 515202, 134004, 2810784, 2810883506682183650, 377198408, 320168

Offset: 2

N. J. A. Sloane, Oct 20 2000

See A057684 for definition.

			For n=3, P=7: trajectory of 7 is 7, 50, 25, 5, 1, 8, 4, 2, 1, 8, 4, 2, 1, 8, 4, 2, 1, ..., which has maximal term 50, cycle length 4 and there are 4 terms before it enters the cycle.

Michael S. Branicky, Table of n, a(n) for n = 2..10001 (terms 2..1000 from T. D. Noe)

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057690, A057691.

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 11 2023 *)

from sympy import prime, primerange
def a(n):
    P = prime(n)
    x, plst, seen = P, list(primerange(2, P)), set()
    while x > 1 and x not in seen:
        seen.add(x)
        x = next((x//p for p in plst if x%p == 0), P*x+1)
    return max(seen)
print([a(n) for n in range(2, 32)]) # Michael S. Branicky, Dec 11 2023

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000

A057690 Length of cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.

Original entry on oeis.org

3, 3, 4, 4, 3, 4, 4, 5, 4, 6, 3, 4, 4, 6, 5, 5, 3, 4, 6, 3, 6, 5, 5, 4, 4, 5, 6, 4, 4, 8, 5, 4, 5, 5, 5, 3, 4, 6, 4, 6, 4, 8, 3, 5, 6, 4, 7, 5, 4, 5, 7, 4, 6, 4, 6, 6, 6, 3, 12, 4, 5, 5, 6, 3, 4, 4, 4, 5, 5, 4, 7, 6, 4, 5, 9, 5, 3, 4, 4, 6, 3, 8, 4, 6, 5, 6, 3, 5, 6, 6, 8, 5, 5, 6, 7, 5, 5, 4, 3, 4, 5, 5, 5, 5, 4

Offset: 2

N. J. A. Sloane, Oct 20 2000

See A057684 for definition.

Note that not all cycles for the iteration starting with p contain the number 1; a(60), for the prime 281, is the first example of this. Its iterates are: 281, 78962, 39481, 3037, 853398, 426699, 142233, 47411, 6773, 521, 146402, 73201, 1031, 289712, 144856, 72428, 36214, 18107, 953, 267794, 133897, with the last 12 terms cycling. Another example is provided by 2543, the 372nd prime. - T. D. Noe, Apr 02 2008

			For n=4, P=7: trajectory of 7 is 7, 50, 25, 5, 1, 8, 4, 2, 1, 8, 4, 2, 1, 8, 4, 2, 1, ..., which has maximal term 50, cycle length 4 and there are 4 terms before it enters the cycle.

Michel Marcus, Table of n, a(n) for n = 2..10000

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057691.

Cf. A023514.

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 11 2023 *)

f(m, p) = {forprime(q=2, precprime(p-1), if (! (m % q), return (m/q));); m*p+1;}
a(n) = {my(p=prime(n), x=p, list = List()); listput(list, x); while (1, x = f(x, p); for (i=1, #list, if (x == list[i], return (#list - i + 1));); listput(list, x););} \\ Michel Marcus, Jan 12 2021

from sympy import prime, primerange
def a(n):
    P = prime(n)
    x, plst, traj, seen = P, list(primerange(2, P)), [], set()
    while x not in seen:
        traj.append(x)
        seen.add(x)
        x = next((x//p for p in plst if x%p == 0), P*x+1)
    return len(traj) - traj.index(x)
print([a(n) for n in range(2, 107)]) # Michael S. Branicky, Dec 11 2023

a(n) = A023514(n)+1 if the cycle contains the number 1. - Jon Maiga, Jan 12 2021

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000

Corrected by T. D. Noe, Apr 02 2008

A057691 Number of terms before entering cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.

Original entry on oeis.org

5, 13, 4, 10, 25, 11, 68, 14, 39, 34, 9, 4, 5, 5, 16, 16, 234, 23, 16, 5, 11, 5, 63, 116, 18, 18, 33, 288, 47, 29, 317, 14, 12, 61, 60, 6, 16, 10, 5, 14, 46, 5, 6, 15, 105, 4, 11, 48, 44, 5, 6, 10, 5, 55, 15, 14, 25, 17, 9, 16, 6, 7, 26, 5, 33, 46, 5, 16, 23, 13, 15, 11, 16, 14, 11

Offset: 2

N. J. A. Sloane, Oct 20 2000

See A057684 for definition.

			For n=3, P=7: trajectory of 7 is 7, 50, 25, 5, 1, 8, 4, 2, 1, 8, 4, 2, 1, 8, 4, 2, 1, ..., which has maximal term 50, cycle length 4 and there are 4 terms before it enters the cycle.

Michael S. Branicky, Table of n, a(n) for n = 2..10001 (terms 2..1000 from T. D. Noe)

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057690.

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 11 2023 *)

from sympy import prime, primerange
def a(n):
    P = prime(n)
    x, plst, traj, seen = P, list(primerange(2, P)), [], set()
    while x not in seen:
        traj.append(x)
        seen.add(x)
        x = next((x//p for p in plst if x%p == 0), P*x+1)
    return traj.index(x)
print([a(n) for n in range(2, 82)]) # Michael S. Branicky, Dec 11 2023

More terms from Larry Reeves (larryr(AT)acm.org), Nov 08 2000

A057684 Trajectory of 13 under the '13x+1' map.

Original entry on oeis.org

13, 170, 85, 17, 222, 111, 37, 482, 241, 3134, 1567, 20372, 10186, 5093, 463, 6020, 3010, 1505, 301, 43, 560, 280, 140, 70, 35, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7, 1, 14, 7

Offset: 0

N. J. A. Sloane, Oct 20 2000

The 'Px+1 map': if x is divisible by any prime < P then divide out these primes one at a time starting with the smallest; otherwise multiply x by P and add 1.

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

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057685, A057686, A057687, A057689, A057690, A057691.

with(numtheory): a := proc(n,S,Q) option remember: local k; if n=0 then RETURN(S); fi: for k from 1 to Q do if a(n-1,S,Q) mod ithprime(k) = 0 then RETURN(a(n-1,S,Q)/ithprime(k)); fi: od: RETURN(ithprime(Q+1)*a(n-1,S,Q)+1) end; # run with S=13 and Q=5.

a[n_, S_, Q_] := a[n, S, Q] = Module[{k}, If[n == 0, S, For[k = 1, k <= Q, k++, If[Mod[a[n-1, S, Q], Prime[k]] == 0, Return[a[n-1, S, Q]/Prime[k]]] ]; Prime[Q+1]*a[n-1, S, Q] + 1]];
Table[a[n, 13, 5], {n, 0, 60}] (* Jean-François Alcover, Jul 13 2016, adapted from Maple *)

A057685 Trajectory of 19 under the `19x+1' map.

Original entry on oeis.org

19, 362, 181, 3440, 1720, 860, 430, 215, 43, 818, 409, 7772, 3886, 1943, 36918, 18459, 6153, 2051, 293, 5568, 2784, 1392, 696, 348, 174, 87, 29, 552, 276, 138, 69, 23, 438, 219, 73, 1388, 694, 347, 6594, 3297, 1099, 157, 2984, 1492, 746

Offset: 0

N. J. A. Sloane, Oct 20 2000

See A057684 for definition.

T. D. Noe, Table of n, a(n) for n=0..72
Index entries for linear recurrences with constant coefficients, signature (0,0,0,1).

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057690, A057691.

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 10 2023 *)

A057686 Trajectory of 23 under the `23x+1' map.

Original entry on oeis.org

23, 530, 265, 53, 1220, 610, 305, 61, 1404, 702, 351, 117, 39, 13, 1, 24, 12, 6, 3, 1, 24, 12, 6, 3, 1, 24, 12, 6, 3, 1, 24, 12, 6, 3, 1, 24, 12, 6, 3, 1, 24, 12, 6, 3, 1, 24, 12, 6, 3, 1, 24, 12, 6, 3, 1, 24, 12, 6, 3, 1, 24, 12, 6, 3, 1, 24, 12, 6, 3, 1

Offset: 0

N. J. A. Sloane, Oct 20 2000

See A057684 for definition.

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

Cf. A057446, A057216, A057522, A057534, A057614.

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 10 2023 *)

A057687 Trajectory of 29 under the `29x+1' map.

Original entry on oeis.org

29, 842, 421, 12210, 6105, 2035, 407, 37, 1074, 537, 179, 5192, 2596, 1298, 649, 59, 1712, 856, 428, 214, 107, 3104, 1552, 776, 388, 194, 97, 2814, 1407, 469, 67, 1944, 972, 486, 243, 81, 27, 9, 3, 1, 30, 15, 5, 1, 30, 15, 5, 1, 30, 15

Offset: 0

N. J. A. Sloane, Oct 20 2000

See A057684 for definition.

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

Cf. A057446, A057216, A057522, A057534, A057614. See also A033478, A057688, A057684, A057685, A057686, A057687, A057689, A057690, A057691.

Px1[p_,n_]:=Catch[For[i=1,iPaolo Xausa, Dec 10 2023 *)

A232711 Conjectured list of numbers whose trajectory under the '5x+1' map eventually reaches 1.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 15, 16, 19, 24, 30, 32, 38, 48, 51, 60, 64, 65, 76, 96, 97, 102, 120, 128, 130, 137, 152, 155, 163, 175, 192, 194, 204, 219, 240, 243, 256, 260, 274, 304, 307, 310, 326, 343, 350, 384, 388, 397, 408, 417, 429, 438, 480, 486, 491, 512

Offset: 1

Jon Perry, Nov 28 2013

nonn

This is conjectural in that there is no known proof that 7, 9, 11, etc. (see A267970) do not eventually cycle. - N. J. A. Sloane, Jan 23 2016
It appears that most numbers diverge, but nothing is known for certain.
Note that the computer programs do not actually calculate a complete list of "numbers k such that the Collatz-like map T: if x odd, x -> 5*x+1 and if x even, x -> x/2, when started at k, eventually reaches 1".

			Beginning with 15 we get the trajectory 15, 76, 38, 19, 96, 48, 24, 12, 6, 3, 16, 8, 4, 2, 1, so 15 is a term.

Dmitry Kamenetsky, Table of n, a(n) for n = 1..1000
Tomás Oliveira e Silva, The px+1 problem.
Index entries for sequences related to 3x+1 (or Collatz) problem

See A267969, A267970 for other trajectories under this map T.
Cf. A057688, A133419.
Cf. A070165 (usual Collatz iteration).

for (i=1;i<2000;i++) {
c=0;
n=i;
while (n>1) {c++;if (n%2==0) n/=2; else n=5*n+1;if (c>100) break;}
if (c<101) document.write(i+", ");
} (Warning: bad program - will not find all the terms. - N. J. A. Sloane, Jan 23 2016)

cli=Compile[{{n,Integer}},If[OddQ[n],5n+1,n/2]//Round,RuntimeAttributes->{Listable},Parallelization->True]; okQ[n]:=Length[NestWhileList[cli,n, #>1&, 1,200]]<200; Select[Range[550],okQ] (* Harvey P. Dale, May 28 2014 *) (Warning: bad program - will not find all the terms. - N. J. A. Sloane, Jan 23 2016)

Entry revised (corrected definition, added warnings to programs, deleted b-file) by N. J. A. Sloane, Jan 23 2016

A368085 Square array read by ascending antidiagonals: row n is the trajectory of P under the 'Px+1' map, where P = n-th prime.

Original entry on oeis.org

2, 3, 5, 5, 10, 11, 7, 26, 5, 23, 11, 50, 13, 16, 47, 13, 122, 25, 66, 8, 95, 17, 170, 61, 5, 33, 4, 191, 19, 290, 85, 672, 1, 11, 2, 383, 23, 362, 145, 17, 336, 8, 56, 1, 767, 29, 530, 181, 29, 222, 168, 4, 28, 4, 1535, 31, 842, 265, 3440, 494, 111, 84, 2, 14, 2, 3071

Offset: 1

Paolo Xausa, Dec 11 2023

The 'Px+1 map' is defined as follows: if there exists p = smallest prime < P which divides x then x = x/p, otherwise x = P*x + 1.

			Array begins:
  [ 1]   2,   5,  11,    23,   47,   95, 191, 383,  767, ... = A153893
  [ 2]   3,  10,   5,    16,    8,    4,   2,   1,    4, ... = A033478
  [ 3]   5,  26,  13,    66,   33,   11,  56,  28,   14, ... = A057688
  [ 4]   7,  50,  25,     5,    1,    8,   4,   2,    1, ... = A368113
  [ 5]  11, 122,  61,   672,  336,  168,  84,  42,   21, ... = A368114
  [ 6]  13, 170,  85,    17,  222,  111,  37, 482,  241, ... = A057684
  [ 7]  17, 290, 145,    29,  494,  247,  19, 324,  162, ... = A368115
  [ 8]  19, 362, 181,  3440, 1720,  860, 430, 215,   43, ... = A057685
  [ 9]  23, 530, 265,    53, 1220,  610, 305,  61, 1404, ... = A057686
  [10]  29, 842, 421, 12210, 6105, 2035, 407,  37, 1074, ... = A057687
  ...    |    |    |
      A000040 | A066885 (from n = 2)
           A066872

Paolo Xausa, Table of n, a(n) for n = 1..11325 (antidiagonals 1..150, flattened).

Rows 1-10: A153893, A033478, A057688, A368113, A368114, A057684, A368115, A057685, A057686, A057687.
Columns 1-3: A000040, A066872, A066885 (from n = 2).
Main diagonal gives A368159.
Cf. A057689, A057690, A057691.

Px1[p_,n_]:=Catch[For[i=1,iA368085list[dmax_]:=With[{a=Reverse[Table[NestList[Px1[Prime[n],#]&,Prime[n],dmax-n],{n,dmax}]]},Array[Diagonal[a,#]&,dmax,1-dmax]];
A368085list[15] (* Generates 15 antidiagonals *)

A259193 5x + 1 sequence beginning at 11.

Original entry on oeis.org

11, 56, 28, 14, 7, 36, 18, 9, 46, 23, 116, 58, 29, 146, 73, 366, 183, 916, 458, 229, 1146, 573, 2866, 1433, 7166, 3583, 17916, 8958, 4479, 22396, 11198, 5599, 27996, 13998, 6999, 34996, 17498, 8749, 43746, 21873, 109366, 54683, 273416, 136708, 68354, 34177, 170886, 85443

Offset: 0

Alonso del Arte, Jun 21 2015

			11 is odd, so it's followed by 5 * 11 + 1 = 56.
56 is even, so it's followed by 56/2 = 28.

Alonso del Arte, Table of n, a(n) for n = 0..999
Alex V. Kontorovich & Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems arXiv:0910.1944 [math.NT], 2009.

Cf. A259207, A057688.

[n eq 1 select 11 else IsOdd(Self(n-1)) select 5*Self(n-1)+1 else Self(n-1) div 2: n in [1..80]]; // Vincenzo Librandi, Jul 04 2015

NestList[If[EvenQ[#], #/2, 5# + 1] &, 5, 100]

a(0) = 11; a(n) = 5a(n - 1) + 1 if a(n - 1) is odd, a(n) = a(n - 1)/2 otherwise.

cp's OEIS Frontend

A057689 Maximal term in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if no such term exists.

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A057690 Length of cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.

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A057691 Number of terms before entering cycle in trajectory of P under the 'Px+1' map, where P = n-th prime, or -1 if trajectory does not cycle.

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A057684 Trajectory of 13 under the '13x+1' map.

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Maple

Mathematica

A057685 Trajectory of 19 under the `19x+1' map.

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Mathematica

A057686 Trajectory of 23 under the `23x+1' map.

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Mathematica

A057687 Trajectory of 29 under the `29x+1' map.

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Mathematica

A232711 Conjectured list of numbers whose trajectory under the '5x+1' map eventually reaches 1.

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