A025586 -id:A025586 - cp's OEIS frontend

A095387 Initial values for 3x+1 trajectories of which the largest term is 9232.

27, 31, 41, 47, 54, 55, 62, 63, 71, 73, 82, 83, 91, 94, 95, 97, 103, 107, 108, 109, 110, 111, 121, 124, 125, 126, 129, 137, 142, 143, 145, 146, 147, 155, 159, 161, 164, 165, 166, 167, 171, 175, 182, 183, 188, 189, 190, 193, 194, 195, 199, 206, 207, 214, 215

Offset: 1

Labos Elemer, Jun 14 2004

Altogether 1579 such initial values exist: 27, ..., 9232.

Cf. A025586.

c[x_]:=c[x]=(1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1);c[1]=1; fpl[x_]:=Delete[FixedPointList[c, x], -1] {ta=Table[0, {1580}], u=1}; Do[If[Equal[Max[fpl[n]], 9232], ta[[u]]=n;u=u+1], {n, 1, 9232}];ta

1579 solutions to A025586(x) = 9232.

A348006 Largest increment in the trajectory from n to 1 in the Collatz map (or 3x+1 problem), or -1 if no such trajectory exists.

Original entry on oeis.org

0, 0, 11, 0, 11, 11, 35, 0, 35, 11, 35, 11, 27, 35, 107, 0, 35, 35, 59, 11, 43, 35, 107, 11, 59, 27, 6155, 35, 59, 107, 6155, 0, 67, 35, 107, 35, 75, 59, 203, 11, 6155, 43, 131, 35, 91, 107, 6155, 11, 99, 59, 155, 27, 107, 6155, 6155, 35, 131, 59, 203, 107

Offset: 1

Paolo Xausa, Oct 02 2021

The largest increment occurs when the trajectory reaches its largest value via a 3x+1 step.

All nonzero terms are odd, since they are of the form 2k+1, for some k >= 5.

			a(3) = 11 because the trajectory starting at 3 is 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1, and the largest increment (from 5 to 16) is 11.
a(4) = 0 because there are only halving steps in the Collatz trajectory starting at 4.

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

Cf. A006370, A025586, A070165, A087232.

nterms=100;Table[c=n;mr=0;While[c>1,If[OddQ[c],mr=Max[mr,2c+1];c=3c+1,c/=2^IntegerExponent[c,2]]];mr,{n,nterms}]

a(n)=n>>=valuation(n,2); my(r); while(n>1, my(t=2*n+1); n+=t; n>>=valuation(n,2); if(t>r, r=t)); r \\ Charles R Greathouse IV, Oct 25 2022

def A348006(n):
    c, mr = n, 0
    while c > 1:
        if c % 2:
            mr = max(mr, 2*c+1)
            c = 3*c+1
        else:
            c //= 2
    return mr
print([A348006(n) for n in range(1, 100)])

If n = 2^k (for k >= 0), a(n) = 0; otherwise a(n) = 2*A087232(n)+1 = (2*A025586(n)+1)/3 = A025586(n)-A087232(n).

A350369 a(n) is the length of the longest sequence of consecutive tripling steps in the Collatz (3x+1) sequence beginning at n.

Original entry on oeis.org

0, 0, 2, 0, 1, 2, 3, 0, 3, 1, 2, 2, 1, 3, 4, 0, 1, 3, 2, 1, 1, 2, 3, 2, 2, 1, 6, 3, 2, 4, 6, 0, 2, 1, 2, 3, 3, 2, 3, 1, 6, 1, 3, 2, 1, 3, 6, 2, 3, 2, 2, 1, 1, 6, 6, 3, 3, 2, 2, 4, 3, 6, 6, 0, 3, 2, 2, 1, 1, 2, 6, 3, 6, 3, 2, 2, 2, 3, 4, 1, 3, 6, 6, 1, 1, 3, 3

Offset: 1

Kevin P. Thompson, Dec 27 2021

"Consecutive tripling steps" are repeated (3x+1)/2 operations that are not interrupted by a second division by 2.
This sequence attempts to measure the largest upward thrust in each Collatz sequence and so is correlated to some degree with the maximum value (A025586) and length (A006577) of Collatz sequences.
If n = 2^x * (2^y*z - 1), then a(n) >= y. - Charles R Greathouse IV, Oct 25 2022

			The Collatz sequence for n=7 has a streak of 3 consecutive tripling steps (at 7, 11, and 17), so a(7) = 3.
7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1
^      ^       ^

Kevin P. Thompson, Table of n, a(n) for n = 1..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A006577, A006667, A025586, A213215, A221469, A347270, A347409.

a(n)=my(c,r); n>>=valuation(n,2); while(n>1, n+=(n+1)/2; if(n%2, c++, r=max(r,c+1); n>>=valuation(n,2); c=0)); max(r,c) \\ Charles R Greathouse IV, Oct 25 2022

A380138 a(n) is the largest value in the '3x+1' trajectory of starting points producing a record number of steps.

Original entry on oeis.org

1, 2, 16, 16, 52, 52, 52, 88, 9232, 9232, 9232, 9232, 9232, 9232, 9232, 9232, 9232, 9232, 250504, 190996, 190996, 250504, 250504, 250504, 481624, 975400, 975400, 497176, 11003416, 11003416, 106358020, 18976192, 41163712, 106358020, 21933016, 104674192, 593279152

Offset: 1

Hugo Pfoertner, Jan 13 2025

nonn

Hugo Pfoertner, Table of n, a(n) for n = 1..148

Cf. A006877, A006878, A006884, A006885, A025586.

s = Map[ToExpression,
  StringSplit[
    Import["https://oeis.org/A006877/b006877.txt", "Data"][[2 ;; -1]]
  ][[All, -1]] ];
Map[Max@ NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, #, # > 1 &] &, s] (* Michael De Vlieger, Jan 13 2025 *)

a(n) = A025586(A006877(n)).

A087262 Integer quotient of largest and initial values in 3x+1 iteration, started at n.

Original entry on oeis.org

1, 1, 5, 1, 3, 2, 7, 1, 5, 1, 4, 1, 3, 3, 10, 1, 3, 2, 4, 1, 3, 2, 6, 1, 3, 1, 341, 1, 3, 5, 297, 1, 3, 1, 4, 1, 3, 2, 7, 1, 225, 1, 4, 1, 3, 3, 196, 1, 3, 1, 4, 1, 3, 170, 167, 1, 3, 1, 5, 2, 3, 148, 146, 1, 3, 1, 4, 1, 3, 2, 130, 1, 126, 1, 4, 1, 3, 3, 10, 1, 3, 112, 111, 1, 3, 2, 6, 1, 3, 1, 101

Offset: 1

Labos Elemer, Sep 11 2003

nonn

Remarkably often, several consecutive terms are identical or close, showing closeness of peaks too: at n=107-111, a(n)=83-86.
If a(n)=1, then the peak is the start-value (per A166245).
It is conjectured that if peak/initial value is an integer then it equals 1.

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

Cf. A025586, A056959, A166245 (indices of 1's).

c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1)c[1]=1; fpl[x_] := Delete[FixedPointList[c, x], -1] Table[Floor[Max[fpl[w]]/w//N], {w, 1, 256}]

a(n) = floor(A025586(n)/n).

A087263 a(n) is the least initial value of a 3x+1 trajectory in which n is the largest (peak) term or a(n) = 0 if n cannot be a peak value (i.e., when n = 2k+1, n = 4k+2, n = 16k+12, etc.).

Original entry on oeis.org

1, 2, 0, 4, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 3, 0, 0, 0, 20, 0, 0, 0, 24, 0, 0, 0, 0, 0, 0, 0, 32, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 48, 0, 0, 0, 7, 0, 0, 0, 56, 0, 0, 0, 0, 0, 0, 0, 21, 0, 0, 0, 68, 0, 0, 0, 72, 0, 0, 0, 0, 0, 0, 0, 80, 0, 0, 0, 84, 0, 0, 0, 19, 0, 0, 0, 0, 0, 0, 0, 96, 0, 0, 0

Offset: 1

Labos Elemer, Sep 11 2003

nonn

Cf. A025586.

c[x_] := (1-Mod[x, 2])*(x/2)+Mod[x, 2]*(3*x+1); c[1]=1; fpl[x_] := Max[Delete[FixedPointList[c, x], -1]] t=Table[fpl[w], {w, 1, 15000}]; Table[Min[Flatten[Position[t, j]]], {j, 1, 256}]

A087970 Maximal term in Collatz-iteration started at 2^n+1.

Original entry on oeis.org

2, 16, 16, 52, 52, 100, 196, 9232, 9232, 1540, 3076, 9232, 12292, 24580, 49156, 98308, 196612, 393220, 786436, 1572868, 3145732, 6291460, 12582916, 25165828, 50331652, 100663300, 201326596, 402653188, 805306372, 1610612740, 3221225476, 6442450948, 12884901892

Offset: 0

Labos Elemer, Sep 24 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 0..1000

Cf. A000051, A025586, A087701, A087972.

a(n) = A025586(2^n+1).

Offset changed to 0, a(0) prepended and more terms added by Amiram Eldar, Jun 04 2024

A087971 Maximal term in Collatz-iteration started at 3^n-1.

Original entry on oeis.org

2, 8, 40, 80, 9232, 9232, 3280, 6560, 29524, 59048, 1276936, 1276936, 6810136, 6810136, 21523360, 43046720, 1570824736, 1570824736, 1743392200, 3486784400, 17651846032, 31381059608, 141214768240, 282429536480, 9161049517720

Offset: 1

Labos Elemer, Sep 24 2003

nonn

Amiram Eldar, Table of n, a(n) for n = 1..1000

Cf. A025586, A087701, A087972.

a(n) = A025586(3^n-1).

A087973 Maximal term in Collatz-iteration started at 3^n.

Original entry on oeis.org

16, 52, 9232, 244, 9232, 2464, 10528, 19684, 88576, 2270104, 1008916, 1594324, 7174456, 65451076, 64570084, 129140164, 1570824736, 1961316232, 8825923024, 10460353204, 47071589416, 105911076184, 423644304724, 66034034786644

Offset: 1

Labos Elemer, Sep 24 2003

nonn

			Compare to A087972, when iv=1+3^n.

Cf. A025586, A087701, A087972.

Table[Max[NestList[If[EvenQ[#],#/2,3#+1]&,3^n,100]],{n,25}] (* Harvey P. Dale, Dec 08 2011 *)

a(n)=A025586[3^n]=A025586[1+3^(n+1)]=A087972[n+1]

A225105 Odd numbers n such that the largest odd term in Collatz(3x+1) trajectory of n is prime.

Original entry on oeis.org

3, 5, 7, 9, 11, 13, 15, 17, 19, 23, 25, 29, 35, 37, 39, 53, 59, 61, 67, 75, 79, 87, 89, 99, 101, 105, 113, 115, 119, 131, 149, 153, 157, 173, 179, 181, 187, 197, 211, 219, 229, 241, 247, 249, 255, 267, 269, 277, 281, 307, 317, 329, 349, 371, 373, 383, 397

Offset: 1

Jayanta Basu, Apr 28 2013

nonn

			15 is in the list since highest odd number in Collatz trajectory of 15 is 53, a prime.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A087232.

Cf. A010051, A070165, A005408, A220237, A025586.

a225105 n = a225105_list !! (n-1)
a225105_list = filter
   ((== 1) . a010051' . maximum . filter odd . a070165_row) a005408_list
-- Reinhard Zumkeller, Apr 30 2013

Coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3*# + 1] &,n, #>1&];t={};Do[If[PrimeQ[Max[Select[Coll[n],OddQ]]],AppendTo[t,n]],{n,1,300,2}];t

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A095387 Initial values for 3x+1 trajectories of which the largest term is 9232.

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A348006 Largest increment in the trajectory from n to 1 in the Collatz map (or 3x+1 problem), or -1 if no such trajectory exists.

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A350369 a(n) is the length of the longest sequence of consecutive tripling steps in the Collatz (3x+1) sequence beginning at n.

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A380138 a(n) is the largest value in the '3x+1' trajectory of starting points producing a record number of steps.

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A087262 Integer quotient of largest and initial values in 3x+1 iteration, started at n.

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A087263 a(n) is the least initial value of a 3x+1 trajectory in which n is the largest (peak) term or a(n) = 0 if n cannot be a peak value (i.e., when n = 2k+1, n = 4k+2, n = 16k+12, etc.).

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A087970 Maximal term in Collatz-iteration started at 2^n+1.

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A087971 Maximal term in Collatz-iteration started at 3^n-1.

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A087973 Maximal term in Collatz-iteration started at 3^n.

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