A267970 -id:A267970 - cp's OEIS frontend

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

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

A267969 Conjectured list of positive numbers k such that the Collatz-like map T: if x > 1 and x odd, x -> 5*x+1 and, if x even, x -> x/2, when started at k, eventually reaches a cycle that does not contain 1 (cf. A232711).

Original entry on oeis.org

5, 10, 13, 17, 20, 26, 27, 33, 34, 40, 43, 52, 54, 66, 68, 80, 83, 86, 104, 105, 108, 132, 136, 160, 166, 172, 181, 185, 208, 210, 211, 215, 216, 245, 263, 264, 269, 272, 275, 320, 329, 332, 344, 362, 370, 416, 420, 422, 430, 432, 435, 453, 457, 463, 490, 526

Offset: 1

Michel Lagneau, Jan 19 2016

nonn

A companion to A232711 (which conjecturally lists the numbers whose trajectory reaches 1), and A267970 (which conjecturally lists the numbers whose trajectory diverges).
This is conjectural in that there is no proof that the list is complete as far as it goes. Some of the terms in A267970 could belong to A232711 or this sequence if the trajectory is extended far enough. - N. J. A. Sloane, Jan 23 2016
It appears that the trajectories of all terms in this sequence reach one of two length 10 loops, one containing 13 and the other 17. This has been checked for terms up to 10^4 assuming trajectories with more than 10000 odd terms are infinite and also up to 10^6 assuming trajectories with more than 1000 odd terms are infinite. - Gary Detlefs, Jan 25 2022

Dmitry Kamenetsky, Table of n, a(n) for n = 1..2051

Cf. A232711, A267970.

f:= proc(m,b,n) if n mod 2 = 1 then return m*n+1 else return n/2 fi end proc
F:= proc(m,b,n,i) option remember; if i=1 then return f(m,b,n) else return f(m,b,F(m,b,n,i-1)) fi end proc
for x from 1 to 1000 do for y from 1 to 1000 do if F(5,1,x,y)= 86 or F(5,1,x,y)=26 then print(x): x=x+1; y:=y+1 fi od od
# use print(x,y) to give the number of iterations needed to reach the cycle point
# Gary Detlefs, Jan 25 2022

Entry revised by N. J. A. Sloane, Jan 23 2016

a(16)-a(55) added by Gary Detlefs, Jan 25 2022

A267703 Conjectured list of numbers whose trajectory under the '7x+1' map eventually reaches 1.

Original entry on oeis.org

1, 2, 4, 5, 8, 9, 10, 16, 18, 20, 32, 36, 40, 41, 64, 72, 73, 80, 82, 128, 144, 146, 160, 164, 167, 256, 288, 292, 320, 328, 329, 334, 512, 576, 584, 585, 640, 656, 658, 668, 1024, 1152, 1168, 1170, 1280, 1312, 1316, 1336, 1337, 1965, 2048, 2304, 2336, 2340, 2560

Offset: 1

Michel Lagneau, Jan 19 2016

nonn

This is conjectural in that there is no known proof that the missing numbers 3, 6, 7, ... are really missing. It may be that after a very large number of iterations they will cycle. - N. J. A. Sloane, Jan 23 2016

Note that the computer program does not actually calculate a complete list of "numbers k such that the Collatz-like map T: if x odd, x -> 7*x+1 and if x even, x -> x/2, when started at k, eventually reaches 1".

			5 is in the sequence because the trajectory of 5 is 5 -> 36 -> 18 -> 9 -> 64 -> 32 -> 16 -> 8 -> 4 -> 2 -> 1.

Dmitry Kamenetsky, Table of n, a(n) for n = 1..164

Cf. A000079, A006577, A023001, A232711, A267969, A267970.

nn:=10000:
for n from 1 to 2340 do:
  m:=n:cyc:={n}:
    for i from 1 to nn do:
     if irem(m,2)=0
      then
       m:=m/2:
      else
      m:=7*m+1:
     fi:
    cyc:=cyc union {m}:
    od:
    n0:=nops(cyc):
    if n0N. J. A. Sloane, Jan 23 2016)

Entry revised by N. J. A. Sloane, Jan 23 2016

a(19)-a(55) from Dmitry Kamenetsky, Jun 24 2024

A268338 Numbers that cycle under the following transformation: if m is even, divide by 2, if m is congruent to 1 mod 4, multiply by 3 and add 1; if m is congruent to 3 mod 4, multiply by 7 and add 1.

Original entry on oeis.org

1, 2, 4, 19, 23, 31, 38, 41, 46

Offset: 1

David Consiglio, Jr., Feb 01 2016

Some numbers appear to grow indefinitely under these rules, but it is possible that they may eventually cycle at some point. All numbers up to 50 either cycle or transform to another number that cycles (typically 1). 51 is the first open case: it may eventually cycle or may continue to grow indefinitely.

			23 is a member of this sequence. 23 is congruent to 3 mod 4.  As a result, 23 transforms to 23*7+1 = 162.  From there 162 -> 81 -> 244 -> 122 -> 61 -> 184 -> 92 -> 46 -> 23.  23 is the least member of this cycle.
49 is not a member of this sequence because it eventually reduces to 19, which cycles.

Cf. A267703, A267970, A267969, A006666, A006370, A005186, A267970, A232711.

a = 1
b = 1
prev = []
keep = []
count = 0
while b < 51:
    keep.append(a)
    flag1 = False
    flag2 = False
    if a % 2 == 0:
        a /= 2
    elif a % 4 == 1:
        a = a*3+1
    else:
        a = a*7+1
    if count > 50:
        b += 1
        a = b
        count = 0
        keep = []
    if keep.count(a) == 2 and a not in prev and a <= 50:
        prev.append(a)
        count = 0
        keep = []
        b += 1
        a = b
    count += 1
print(sorted(prev))
# David Consiglio, Jr., Feb 01 2016

Corrected and edited by David Consiglio, Jr., Apr 20 2016

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

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A267969 Conjectured list of positive numbers k such that the Collatz-like map T: if x > 1 and x odd, x -> 5*x+1 and, if x even, x -> x/2, when started at k, eventually reaches a cycle that does not contain 1 (cf. A232711).

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A267703 Conjectured list of numbers whose trajectory under the '7x+1' map eventually reaches 1.

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A268338 Numbers that cycle under the following transformation: if m is even, divide by 2, if m is congruent to 1 mod 4, multiply by 3 and add 1; if m is congruent to 3 mod 4, multiply by 7 and add 1.

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