A060412 -id:A060412 - cp's OEIS frontend

A161021 Collatz (or 3x+1) trajectory starting at 703.

703, 2110, 1055, 3166, 1583, 4750, 2375, 7126, 3563, 10690, 5345, 16036, 8018, 4009, 12028, 6014, 3007, 9022, 4511, 13534, 6767, 20302, 10151, 30454, 15227, 45682, 22841, 68524, 34262, 17131, 51394, 25697, 77092, 38546, 19273, 57820, 28910

Offset: 0

Reinhard Zumkeller, Jun 03 2009

703 = A060412(5); a(A006577(703)) = a(170) = 1;
a(n) = A008884(n-59) for n with 162 <= n <= 170;
a(n) = A161022(n+53) for n with 155 <= n <= 170;
a(n) = A161023(n+153) for n with 144 <= n <= 170.

R. Zumkeller, Table of n, a(n) for n = 0..170
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

NestList[If[EvenQ[#],#/2,3#+1]&,703,40] (* Harvey P. Dale, Nov 27 2011 *)

Edited by N. J. A. Sloane, Jul 27 2019

A161022 Collatz (or 3x+1) trajectory starting at 10087.

Original entry on oeis.org

10087, 30262, 15131, 45394, 22697, 68092, 34046, 17023, 51070, 25535, 76606, 38303, 114910, 57455, 172366, 86183, 258550, 129275, 387826, 193913, 581740, 290870, 145435, 436306, 218153, 654460, 327230, 163615, 490846, 245423, 736270

Offset: 0

Reinhard Zumkeller, Jun 03 2009

10087 = A060412(6); a(A006577(10087)) = a(223) = 1;
a(n)=A161021(n-53)=A161023(n+100) for n with 208<=n<=228.
At step 221 becomes periodic: 4 2 1 4 2 1 4 2 1 ... - N. J. A. Sloane, Jul 27 2019

R. Zumkeller, Table of n, a(n) for n = 0..223
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

A008884.

NestList[If[EvenQ[#],#/2,3#+1]&,10087,30] (* Harvey P. Dale, Jul 25 2023 *)

Edited by N. J. A. Sloane, Jul 27 2019

A161023 Collatz trajectory starting at 35655.

Original entry on oeis.org

35655, 106966, 53483, 160450, 80225, 240676, 120338, 60169, 180508, 90254, 45127, 135382, 67691, 203074, 101537, 304612, 152306, 76153, 228460, 114230, 57115, 171346, 85673, 257020, 128510, 64255, 192766, 96383, 289150, 144575, 433726

Offset: 0

Reinhard Zumkeller, Jun 03 2009

35655 = A060412(7); a(A006577(35655)) = a(323) = 1.
a(n) = A161021(n-153) for n with 297 <= n <= 323;
a(n) = A161022(n-100) for n with 308 <= n <= 323.
At step 321 becomes periodic: 4 2 1 4 2 1 4 2 1 ... - N. J. A. Sloane, Jul 27 2019

R. Zumkeller, Table of n, a(n) for n = 0..323
Index entries for sequences related to 3x+1 (or Collatz) problem
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A008884.

NestWhileList[If[EvenQ[#],#/2,3#+1]&,35655,#>1&] (* Harvey P. Dale, Apr 28 2016 *)

Edited by N. J. A. Sloane, Jul 27 2019

A217934 Records in A102419.

Original entry on oeis.org

0, 1, 6, 11, 96, 132, 171, 220, 267, 269, 282, 287, 298, 365, 401, 468, 476, 486, 502, 613, 644, 649, 706, 729, 892, 897, 988, 1122, 1161, 1177, 1187, 1445, 1471, 1575, 1614, 1639

Offset: 1

N. J. A. Sloane, Oct 20 2012

If the "1" is omitted, also records in A060445.

			See A102419.

Tomás Oliveira e Silva, Tables
Eric Roosendaal, On the 3x + 1 problem
N. J. A. Sloane, First 36 terms of A217934 and A060412 [From Roosendaal web site]

Cf. A102419, A060445, A060412.

A340418 Indices in A339991 where records occur.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 19, 29, 39, 53, 61, 73, 79, 101, 149, 229

Offset: 1

Ya-Ping Lu, Jan 06 2021

The records in A339991 corresponding the first 16 terms in this sequence are 0, 1, 4, 8, 9, 15, 22, 23, 24, 31, 32, 71, 88, 99, 104, 9267.
A339991(397), which is > 249275, is still unknown.
A339991(397) > 10^6. - Michael S. Branicky, Jan 09 2025

Cf. A339991, A340008, A060412 (Collatz record indices).

With[{s = Array[-1 + Length@ NestWhileList[Which[EvenQ@ #, #/2, PrimeQ@ #, #^2 - 1, True, # - 1] &, #, # > 1 &] &, 396]}, Map[FirstPosition[s, #][[1]] &, Union@ FoldList[Max, s]]] (* Michael De Vlieger, Feb 12 2021 *)

from sympy import isprime
rec = -1
for n in range(1, 397):
    m, ct = n, 0
    while m > 1:
        if m%2 == 0: m //= 2
        elif isprime(m) == 1: m = m*m - 1
        else: m -= 1
        ct += 1
    if ct > rec: print(n); rec = ct

A243115 Starting values of the reduced Collatz function (A014682) where 2 to the power of the "dropping time" is greater than the starting value.

Original entry on oeis.org

3, 7, 11, 15, 23, 27, 31, 39, 47, 59, 63, 71, 79, 91, 95, 103, 111, 123, 127, 155, 159, 167, 175, 191, 199, 207, 219, 223, 231, 239, 251, 255, 283, 287, 303, 319, 327, 347, 359, 367, 383, 411, 415, 423, 447, 463, 479, 487, 495, 507, 511, 539, 543, 559, 575

Offset: 1

K. Spage, Aug 20 2014

nonn

a(n) is the lowest positive starting value of the reduced Collatz function such that all starting values (>1) that are congruent to a(n) (mod 2^d) have the same dropping time (d). The dropping time here counts the (3x+1)/2 and the x/2 steps as listed in A126241. A number is included in this sequence if 2^A126241(a(n)) > a(n).
Starting values that produce new record dropping times as listed in A060412 are necessarily a subset of this sequence.
If at least one iteration is carried out before checking that the absolute iterated value has become less than or equal to the absolute starting value, then a(n) is the lowest positive starting value such that all starting values (positive, zero or negative) that are congruent to a(n) (mod 2^d) have the same dropping time (d). Defined like this, the sequence would start with 0, 1, 3, 7.
For k>0, A076227(k) is the number of terms between 2^k and 2^(k+1)-1. - Ruud H.G. van Tol, Dec 18 2022
All terms are congruent to 3 (mod 4) since any 1 (mod 4) has dropping time A126241(4k+1) = 2, for k>=1. - Ruud H.G. van Tol, Jan 11 2023

			3 is in this sequence because the dropping time starting with 3 is A126241(3) = 4 and 2^4 > 3.

Cf. A014682, A060412, A076227, A126241, A177789, A239126.

is(t)= if(t<3||3!=t%4,0,my(x=t, d=0); until(x<=t, if(x%2, x=(x*3+1)/2, x/=2); d++); 2^d>t); \\ updated by Ruud H.G. van Tol, Jan 10 2023

Offset 1 from Ruud H.G. van Tol, Jan 10 2023

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A161021 Collatz (or 3x+1) trajectory starting at 703.

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A161022 Collatz (or 3x+1) trajectory starting at 10087.

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A161023 Collatz trajectory starting at 35655.

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A217934 Records in A102419.

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A340418 Indices in A339991 where records occur.

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A243115 Starting values of the reduced Collatz function (A014682) where 2 to the power of the "dropping time" is greater than the starting value.

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