A070165 -id:A070165 - cp's OEIS frontend

A347668 Indices of records in A347409.

1, 2, 3, 15, 21, 75, 151, 1365, 5461, 7407, 14563, 87381, 184111, 932067, 5592405, 13256071, 26512143, 357913941, 1431655765, 3817748707, 22906492245, 91625968981, 244335917283, 1466015503701, 5212499568715, 10424999137431, 93824992236885

Offset: 1

Paolo Xausa, Sep 10 2021

Conjecture 1: A347409(a(n)) is even for n >= 11. Conjecture 2: all even numbers > 2 appear as A347409(a(n)) for some n. - Chai Wah Wu, Sep 29 2021

If conjectures 1 and 2 are true, then A347409(a(n)) = 2n - 6 for n >= 11, and hence a(n) <= (4^(n-3)-1)/3 for n >= 11 since A347409((4^(n-3)-1)/3) = 2n - 6. - Charles R Greathouse IV, Oct 25 2022

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

Cf. A006370, A070165, A135282, A347409, A347669, A002450.

A347409[n_]:=(c=n;sm=0;While[c>1,If[OddQ[c],c=3c+1,If[(s=IntegerExponent[c,2])>sm,sm=s];c/=2^s]];sm)
upto=100000;a={};rec=-1;Do[If[(r=A347409[i])>rec,rec=r;AppendTo[a,i]],{i,upto}];a

f(n)=my(nb=0); while (n != 1, if (n % 2, n=3*n+1, my(x = valuation(n, 2)); n /= 2^x; nb = max(nb, x)); ); nb; \\ A347409
lista(nn) = my(r=-1, m); for (n=1, nn, if ((m=f(n)) > r, print1(n, ", "); r = m);); \\ Michel Marcus, Sep 10 2021

a(15) from Michel Marcus, Sep 10 2021
a(16)-a(17) from Alois P. Heinz, Sep 10 2021
a(18)-a(20) from Michael S. Branicky, Sep 28 2021
a(21)-a(22) from Michael S. Branicky, Sep 30 2021
a(23) from Michael S. Branicky, Oct 04 2021
a(24)-a(27) from Kevin P. Thompson, Apr 14 2022

A359247 The bottom entry in the absolute difference triangle of the elements in the Collatz trajectory of n.

Original entry on oeis.org

1, 1, 1, 1, 0, 1, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 1, 1, 0, 0, 0, 1, 1, 1, 1, 0, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 0

Offset: 1

Michel Lagneau, Dec 22 2022

nonn

			a(3) = 1 because the Collatz trajectory of 3 is T = [3, 10, 5, 16, 8, 4, 2, 1], and the absolute difference triangle of the elements of T is:
  3  . 10  .  5  . 16  .  8  .  4  .  2  .  1
     7  .  5  . 11  .  8  .  4  .  2  .  1
        2  .  6  .  3  .  4  .  2  .  1
           4  .  3  .  1  .  2  .  1
              1  .  2  .  1  .  1
                 1  .  1  .  0
                    0  .  1
                       1
with bottom entry a(3) = 1.

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

Cf. A070165, A187203.

Collatz[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&];Flatten[Table[Collatz[n],{n,10}]];Table[d=Collatz[m];While[Length[d]>1,d=Abs[Differences[d]]];d[[1]],{m,100}]

a(n) = my(list=List([n])); while (n!=1, if(n%2, n=3*n+1, n=n/2); listput(list, n)); my(v = Vec(list)); while (#v != 1, v = vector(#v-1, k, abs(v[k+1]-v[k]))); v[1]; \\ Michel Marcus, Dec 23 2022

a(2^n) = 1.

A160000 Number of partitions of n into parts occurring in '3x+1'-trajectory starting with n.

Original entry on oeis.org

1, 2, 3, 4, 5, 11, 9, 10, 17, 20, 23, 66, 35, 57, 51, 36, 77, 153, 122, 126, 61, 213, 189, 883, 353, 338, 337, 851, 645, 570, 571, 202, 1220, 1066, 901, 4017, 3462, 2440, 2777, 1598, 1852, 512, 8084, 4909, 3952, 3122, 3399, 29851, 17813, 11391, 11285, 7128

Offset: 1

Reinhard Zumkeller, May 11 2009

			7-22-11-34-17-52-26-13-40-20-10-5-16-8-[4-2-1]*:
{1,2,4,5,7} is the set of numbers <= n, occurring in this trajectory, therefore a(7) = #{7, 5+2, 5+1+1, 4+2+1, 4+1+1+1, 2+2+2+1, 2+2+1+1+1, 2+1+1+1+1+1, 1+1+1+1+1+1+1} = 9.

R. Zumkeller, Table of n, a(n) for n = 1..100
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A160001, A000041, A159999, A006370.

Cf. A070165.

a160000 n = p (takeWhile (<= n) $ sort $ a070165_row n) n where
   p _          0 = 1
   p []         _ = 0
   p ks'@(k:ks) m = if m < k then 0 else p ks' (m - k) + p ks m
-- Reinhard Zumkeller, Sep 01 2012

A160001 Number of partitions of n into distinct parts occurring in '3x+1'-trajectory starting with n.

Original entry on oeis.org

1, 1, 2, 1, 2, 4, 3, 1, 4, 3, 4, 9, 4, 8, 5, 1, 6, 18, 10, 5, 2, 13, 6, 25, 19, 10, 8, 41, 24, 14, 10, 1, 40, 19, 8, 138, 90, 55, 62, 7, 17, 3, 144, 63, 34, 16, 16, 43, 218, 148, 105, 22, 7, 39, 24, 323, 371, 150, 224, 54, 27, 40, 30, 1, 572, 444, 344, 71, 32, 19, 39, 1766, 52

Offset: 1

Reinhard Zumkeller, May 11 2009

nonn

			7-22-11-34-17-52-26-13-40-20-10-5-16-8-[4-2-1]*:
{1,2,4,5,7} is the set of numbers <= n, occurring in this trajectory, therefore a(7) = #{7, 5+2, 4+2+1} = 3.

R. Zumkeller, Table of n, a(n) for n = 1..100
Wikipedia, Collatz conjecture
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A160000, A000009, A159999, A006370.

Cf. A070165.

a160001 n = p (takeWhile (<= n) $ sort $ a070165_row n) n where
   p _      0 = 1
   p []     _ = 0
   p (k:ks) m = if m < k then 0 else p ks (m - k) + p ks m
-- Reinhard Zumkeller, Sep 01 2012

A174538 The smallest k such that the number of steps in the n Collatz sequences starting at k+i, i=0..n-1, is always prime.

Original entry on oeis.org

3, 3, 3, 60, 246, 560, 560, 560, 4722, 4722, 6032, 6666, 13956, 13956, 13956, 13956, 13956, 13956, 13956, 13956, 13956, 13956, 13956, 13956, 81488, 81488, 81488, 83840, 89535, 89535, 89535, 282880, 282984, 282984, 282984, 282984

Offset: 1

Michel Lagneau, Mar 21 2010

nonn

Indices of A006577 that start a run of at least n primes.

We find long sequences of primes, for example there are 55 primes starting at index 282984, namely 83, 101, 127, 251,...

			A006577(3)=7, A006577(4)=2 and A006577(5)=5 are all prime, so a(1)=a(2)=a(3) =3 mark the start of that run.

Donovan Johnson, Table of n, a(n) for n = 1..619 (terms <= 10^10)
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A008908, A066906, A070165, A033491.

A174538 := proc(n)
        local k,allp;
        for k from 1 do
                allp := true;
                for i from 0 to n-1 do
                        if not isprime(A006577(k+i)) then
                                allp := false;
                                break;
                        end if;
                end do:
                if allp then
                        return k;
                end if;
        end do:
end proc: # R. J. Mathar, Jul 08 2012

Edited by R. J. Mathar, Jul 08 2012

A213181 Number of chains of even numbers of length 2 or more in the Collatz (3x+1) trajectory of n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 3, 1, 4, 1, 3, 2, 2, 3, 2, 1, 3, 4, 4, 2, 1, 3, 2, 2, 5, 2, 17, 4, 4, 2, 16, 1, 6, 3, 2, 5, 4, 4, 6, 2, 17, 1, 6, 4, 4, 2, 16, 2, 5, 5, 5, 3, 2, 17, 17, 4, 7, 4, 6, 3, 3, 16, 15, 1, 6, 6, 5, 4, 3, 2, 16, 5, 18, 4, 2, 5, 5, 6, 6, 2, 4, 17, 17

Offset: 1

Jayanta Basu, Feb 28 2013

nonn

A pair of even numbers that appear side by side in Collatz trajectory of n is considered a chain of length 2 and likewise for chains of greater length.

			For n=3, Collatz trajectory of 3 is 3,10,5,16,8,4,2,1, hence the only chain is 16,8,4,2 and so a(3)=1.
For n=12: 12,6,3,10,5,16,8,4,2,1 and as such there are two chains 12,6 and 16,8,4,2 so a(12)=2.

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

Cf. A014682, A070165, A133872, A139391, A363270.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Table[c = Collatz[n]; cnt = 0; evenCnt = 0; Do[If[OddQ[i], evenCnt = 0, evenCnt++; If[evenCnt == 2, cnt++]], {i, c}]; cnt, {n, 100}] (* T. D. Noe, Feb 28 2013 *)

a(n) = a(A363270(A014682(n))) + 1 for n >= 3. - Alan Michael Gómez Calderón, Apr 09 2025

a(n) = a(A139391(n)) + A133872(n) for n >= 2. - Alan Michael Gómez Calderón, Apr 23 2025

A214614 Irregular triangle read by rows: row n gives numbers <= n whose Collatz trajectory contains the trajectory of n.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 1, 2, 4, 1, 2, 4, 5, 1, 2, 3, 4, 5, 6, 1, 2, 4, 5, 7, 1, 2, 4, 8, 1, 2, 4, 5, 7, 8, 9, 1, 2, 4, 5, 8, 10, 1, 2, 4, 5, 8, 10, 11, 1, 2, 3, 4, 5, 6, 8, 10, 12, 1, 2, 4, 5, 8, 10, 13, 1, 2, 4, 5, 7, 8, 10, 11, 13, 14, 1, 2, 4, 5, 8, 10, 15

Offset: 1

Jayanta Basu, Mar 06 2013

Each row has A159999(n) elements and ends in n.

			Rows of triangle:
{1},
{1, 2},
{1, 2, 3},
{1, 2, 4},
{1, 2, 4, 5},
{1, 2, 3, 4, 5, 6},
{1, 2, 4, 5, 7},
{1, 2, 4, 8},
{1, 2, 4, 5, 7, 8, 9},
{1, 2, 4, 5, 8, 10}

Cf. A070165, A159999.

import Data.List (sort)
a214614 n k = a214614_tabf !! (n-1) (k-1)
a214614_row n = a214614_tabf !! (n-1)
a214614_tabf = zipWith f [1..] a070165_tabf where
                       f v ws = sort $ filter (<= v) ws
-- Reinhard Zumkeller, Sep 01 2014

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; f[n_] := Module[{c = Collatz[n]}, Select[c, # <= n &]]; t = Table[f[n], {n, 20}]; Flatten[t] (* T. D. Noe, Mar 07 2013 *)

A217743 Excess of number of odds in the form 4k+1 over number of odds in the form 4k+3 in Collatz trajectory of n.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 3, 2, 0, 1, 4, 3, 3, 2, 2, 3, 1, 1, 4, 3, -6, 2, 4, 0, -6, 1, 5, 4, 2, 3, 3, 3, 2, 2, -5, 2, 4, 3, 5, 1, -5, 1, 4, 4, 4, 3, 3, -6, -6, 2, 5, 4, 3, 0, 2, -6, -8, 1, 5, 5, 3, 4, 4, 2, -4, 3, -5, 3, 2, 3, 5, 2, 2, 2, 3, -5, -5

Offset: 1

Jayanta Basu, Mar 23 2013

sign

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

Cf. A070165, A192719.

Cf. A217744 (n having equal numbers of 4k+1 and 4k+3 terms).

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3*# + 1] &, n, # > 1 &]; Table[Length[Select[Collatz[n], Mod[#, 4] == 1 &]] - Length[Select[Collatz[n], Mod[#, 4] == 3 &]], {n,60}]
Table[m4=Mod[NestWhileList[If[EvenQ[#],#/2,3*#+1]&,n,#>1&],4];Count[m4, 1]-Count[m4, 3], {n, 60}] (* Zak Seidov, Mar 23 2013 *)

A217744 Numbers whose Collatz trajectory contains equal number of terms of the form 4k+1 and 4k+3.

Original entry on oeis.org

15, 30, 60, 120, 151, 239, 240, 255, 302, 377, 423, 425, 478, 479, 480, 510, 593, 604, 669, 685, 743, 754, 755, 846, 847, 850, 851, 939, 956, 958, 960, 1020, 1053, 1057, 1186, 1191, 1208, 1297, 1321, 1338, 1341, 1370, 1375, 1473, 1486, 1487, 1508, 1509, 1510

Offset: 1

Jayanta Basu, Mar 23 2013

nonn

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

Cf. A070165, A192719, A217743.

Collatz[n_] :=NestWhileList[If[EvenQ[#], #/2, 3*# + 1] &, n, # > 1 &]; t = {}; Do[If[Length[Select[Collatz[n], Mod[#, 4] == 1 &]] -Length[Select[Collatz[n], Mod[#, 4] == 3 &]] == 0, AppendTo[t, n]], {n, 1500}]; t

A219696 Numbers k such that the trajectory of 3k + 1 under the '3x + 1' map reaches k.

Original entry on oeis.org

1, 2, 4, 8, 10, 14, 16, 20, 22, 26, 40, 44, 52, 106, 184, 206, 244, 274, 322, 526, 650, 668, 790, 866, 976, 1154, 1300, 1438, 1732, 1780, 1822, 2308, 2734, 3238, 7288

Offset: 1

Robert C. Lyons, Nov 25 2012

This sequence seems complete; there are no other terms <= 10^9. - T. D. Noe, Dec 03 2012
If the 3x+1 step is replaced with (3x+1)/2, the sequence becomes {1, 2, 4, 8, 10, 14, 20, 22, 26, 40, 44, 206, 244, 650, 668, 866, 1154, 1822, 2308, ...}. - Robert G. Wilson v, Jan 13 2015
From Andrew Slattery, Aug 03 2023: (Start)
For most terms k, the trajectory of 3k + 1 reaches 310 or the trajectory of 310 reaches k.
  For the rest of the terms k, the trajectory of 3k + 1 reaches 22 or the trajectory of 22 reaches k.
  With the exception of k = 1, k is reached after S steps,
   where S = c*8 + d*13 + e*44 + f*75, with c, d, e and f in {0, 1, 2}; in particular, S is in {8, 13, 8+8, 8+13, 13+13, 44, 75, 44+44, 75+13+13, 75+44, 75+75}. (End)

			For k = 4, the Collatz trajectory of 3k + 1 is (13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which includes 4; thus, 4 is in the sequence.
For k = 5, the Collatz trajectory of 3k + 1 is (16, 8, 4, 2, 1), which does not include 5; thus, 5 is not in the sequence.

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

Cf. A014682, A070991, A006370, A070165.

a219696 n = a219696_list !! (n-1)
a219696_list = filter (\x -> collatz'' x == x) [1..] where
   collatz'' x = until (`elem` [1, x]) a006370 (3 * x + 1)
-- Reinhard Zumkeller, Aug 11 2014

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; Select[Range[10000], MemberQ[Collatz[3 # + 1], #] &] (* T. D. Noe, Dec 03 2012 *)

a006370(n) = if(n%2==0, n/2, 3*n+1)
is(n) = my(x=3*n+1); while(1, x=a006370(x); if(x==n, return(1), if(x==1, return(0)))) \\ Felix Fröhlich, Jun 10 2021

def ok(n):
    if n==1: return [1]
    N=3*n + 1
    l=[N, ]
    while True:
        if N%2==1: N = 3*N + 1
        else: N/=2
        l+=[N, ]
        if N<2: break
    if n in l: return 1
    return 0 # Indranil Ghosh, Apr 22 2017

Initial 1 from Clark R. Lyons, Dec 02 2012

cp's OEIS Frontend

A347668 Indices of records in A347409.

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Mathematica

PARI

Extensions

A359247 The bottom entry in the absolute difference triangle of the elements in the Collatz trajectory of n.

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Mathematica

PARI

Formula

A160000 Number of partitions of n into parts occurring in '3x+1'-trajectory starting with n.

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Haskell

A160001 Number of partitions of n into distinct parts occurring in '3x+1'-trajectory starting with n.

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Haskell

A174538 The smallest k such that the number of steps in the n Collatz sequences starting at k+i, i=0..n-1, is always prime.

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Maple

Extensions

A213181 Number of chains of even numbers of length 2 or more in the Collatz (3x+1) trajectory of n.

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Mathematica

Formula

A214614 Irregular triangle read by rows: row n gives numbers <= n whose Collatz trajectory contains the trajectory of n.

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Haskell

Mathematica

A217743 Excess of number of odds in the form 4k+1 over number of odds in the form 4k+3 in Collatz trajectory of n.

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