A070165 -id:A070165 - cp's OEIS frontend

A221470 Least m such that the Collatz (3x+1) iteration of m has exactly n increasing peak values.

1, 5, 3, 7, 15, 287, 191, 127, 223, 159, 143, 95, 63, 47, 31, 27, 703, 6471, 6383, 4255, 6887, 4591, 50427, 47867, 31911, 77671, 161439, 113383, 239231, 159487, 1027431, 974079, 730559, 487039, 432923, 288615, 270271, 3041391, 9158655, 6416623, 16786431, 12589823

Offset: 0

T. D. Noe, Jan 17 2013

nonn

Sequence A221469 lists the number of increasing peaks.

			The Collatz iteration starting at 7 is (7, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1), which has 3 increasing peaks: 22, 34, and 52. No number smaller than 7 has 3 increasing peaks. Hence, a(3) = 7.

T. D. Noe, Table of n, a(n) for n = 0..60

Cf. A070165 (Collatz trajectory of n), A221469.

a221470 = (+ 1 ) . fromJust . (`elemIndex` (map a221469 [1..]))
-- Reinhard Zumkeller, Jan 18 2013

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; nn = 20; t = Table[0, {nn}]; found = 0; n = 0; While[found < nn, n++; c = Collatz[n]; cnt = 0; mx = n; Do[If[k > mx, cnt++; mx = k], {k, c}]; If[cnt > 0 && cnt <= nn && t[[cnt]] == 0, t[[cnt]] = n; found++]]; Join[{1}, t]

A223699 Least number whose Collatz (3x+1) iteration has its maximum value at position n (counting from 1).

Original entry on oeis.org

1, 2, 4, 8, 3, 32, 21, 20, 13, 80, 15, 7, 96, 69, 68, 45, 93, 19, 61, 56, 37, 51, 72, 49, 39, 33, 43, 133, 79, 260, 115, 349, 255, 127, 27, 157, 135, 279, 123, 421, 375, 727, 219, 723, 447, 295, 740, 493, 439, 591, 657, 1281, 1159, 877, 759, 615, 519, 1603

Offset: 1

T. D. Noe, Mar 26 2013

nonn

The maximum values are in A223700.

			The Collatz iteration of 15 is {15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1}. The maximum is 160, which occurs at position 11, counting from the right. Hence, a(11) = 15 because no number smaller than 15 has its maximum value at the 11 position.

T. D. Noe, Table of n, a(n) for n = 1..800

Cf. A070165 (Collatz iteration of n), A223700.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; t= Table[c = Reverse[Collatz[n]]; Position[c, Max[c]][[1, 1]], {n, 10000}]; t2 = {}; n = 0; While[n++; p = Position[t, n, 1, 1]; p != {}, AppendTo[t2, p[[1,1]]]]; t2

A223700 The maximum value of the Collatz (3x+1) iteration beginning at A223699(n).

Original entry on oeis.org

1, 2, 4, 8, 16, 32, 64, 20, 40, 80, 160, 52, 96, 208, 68, 136, 280, 88, 184, 56, 112, 232, 72, 148, 304, 100, 196, 400, 808, 260, 520, 1048, 13120, 4372, 9232, 472, 916, 1888, 628, 1264, 2536, 4912, 1672, 3256, 39364, 2248, 740, 1480, 2968, 5992, 1972, 3844

Offset: 1

T. D. Noe, Mar 26 2013

nonn

			The Collatz iteration of 15 is {15, 46, 23, 70, 35, 106, 53, 160, 80, 40, 20, 10, 5, 16, 8, 4, 2, 1}. The maximum is 160, which occurs at position 11, counting from the right. Hence, a(11) = 160 because no number smaller than 15 has its maximum value at the 11 position.

T. D. Noe, Table of n, a(n) for n = 1..800

Cf. A070165 (Collatz iteration of n), A223699.

Collatz[n_] := NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]; t= Table[c = Reverse[Collatz[n]]; Position[c, Max[c]][[1, 1]], {n, 10000}]; t3 = {}; n = 0; While[n++; p = Position[t, n, 1, 1]; p != {}, c = Collatz[p[[1,1]]]; AppendTo[t3, Max[c]]]; t3

A225570 The greedy smallest infinite reverse Collatz (3x+1) sequence.

Original entry on oeis.org

1, 2, 4, 8, 16, 5, 10, 20, 40, 13, 26, 52, 17, 34, 11, 22, 7, 14, 28, 56, 112, 37, 74, 148, 49, 98, 196, 65, 130, 43, 86, 172, 344, 688, 229, 458, 916, 305, 610, 203, 406, 812, 1624, 541, 1082, 2164, 721, 1442, 2884, 961, 1922, 3844, 7688, 15376, 5125, 10250

Offset: 1

David Spies, Jul 29 2013

nonn

For each a(n) (where n > 4), a(n) = (a(n-1) - 1)/3 if the result is an odd integer not divisible by 3. Otherwise a(n) = 2 * a(n-1).
Going backwards from any term a(n) to a(1), this is the Collatz sequence for a(n).  Furthermore, each term in the sequence is the smallest possible term (ignoring multiples of 3) with this property given the previous term.
Multiples of 3 are ignored because after visiting a multiple of 3, subsequent terms can only double.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A014682, A070165.

A225570 := proc(n)
    local a;
    option remember;
    if n <= 4  then
        2^(n-1) ;
    else
        a := (procname(n-1)-1)/3 ;
        if type(a,'integer') and type(a,'odd') and modp(a,3) <> 0 then
            return a;
        else
            return procname(n-1)*2 ;
        end if;
    end if;
end proc: # R. J. Mathar, Aug 03 2013

last = 8; Join[{1, 2, 4, 8}, Table[test = (last - 1)/3; If[OddQ[last] || ! IntegerQ[test] || IntegerQ[test/3], last = 2*last, last = (last - 1)/3]; last, {96}]] (* T. D. Noe, Aug 11 2013 *)

A225840 Largest number less than n occurring in Collatz trajectory starting with n.

Original entry on oeis.org

1, 2, 2, 4, 5, 5, 4, 8, 8, 10, 10, 10, 13, 10, 8, 16, 17, 17, 16, 16, 20, 20, 16, 22, 20, 23, 26, 26, 23, 23, 16, 29, 26, 20, 34, 34, 34, 38, 20, 40, 32, 40, 40, 40, 40, 46, 24, 40, 44, 44, 40, 40, 53, 53, 52, 56, 52, 58, 53, 53, 61, 61, 32, 56, 58, 58, 52

Offset: 2

Reinhard Zumkeller, May 16 2013

nonn

In triangle A070165: a(n)-th row is a suffix of n-th row;
a(n) = A070165(n,k) for some k with 1 <= k < A006577(n);
a(n) <> A070165(m,k) for all k with 1 <= k < A006577(n), a(n) < m < n.

Reinhard Zumkeller, Table of n, a(n) for n = 2..10000
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A187831.

a225840 n = maximum $ filter (< n) $ a070165_row n

scoll[n_]:=Sort[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]]; Flatten[Table[scoll[n][[Flatten[Position[scoll[n],n]-1]]],{n,2,68}]] (* Jayanta Basu, May 28 2013 *)

A226123 Number of terms of the form 2^k in Collatz(3x+1) trajectory of n.

Original entry on oeis.org

1, 2, 5, 3, 5, 5, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 7, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 9, 5, 5, 5, 5, 5, 5, 5, 5, 7, 9, 5, 5

Offset: 1

Jayanta Basu, May 27 2013

nonn

a(n) = sum(A209229(A070165(n,k)): k=1..A006577(n)). - Reinhard Zumkeller, May 30 2013

			a(3)=5 since Collatz trajectory of 3 contains terms 1,2,4,8 and 16.

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

Cf. A070165.

Cf. A135282, A208981.

a226123 = sum . map a209229 . a070165_row
-- Reinhard Zumkeller, May 30 2013

coll[n_]:=NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]; Table[Length[Select[coll[n],IntegerQ[Log[2,#]]&]],{n,87}]

A246436 Number of numbers 1,...,n not occurring in the Collatz trajectory starting with n.

Original entry on oeis.org

0, 0, 0, 1, 1, 0, 2, 4, 2, 4, 4, 3, 6, 4, 8, 11, 8, 4, 8, 12, 15, 10, 14, 13, 11, 16, 17, 12, 15, 19, 21, 26, 16, 22, 26, 16, 19, 21, 21, 31, 28, 34, 23, 28, 31, 34, 34, 36, 29, 29, 33, 40, 43, 36, 40, 36, 33, 39, 37, 44, 47, 45, 48, 57, 42, 41, 45, 53, 56

Offset: 1

Reinhard Zumkeller, Sep 01 2014

nonn

a(n) = n - A159999(n).

			8-4-2-1: a(8) = #{3,5,6,7} = 4;
9-28-14-7-22-11-34-17-52-26-13-40-20-10-5-16-8-4-2-1: a(9) = #{3,6} = 2;
10-5-16-8-4-2-1: a(10) = #{3,6,7,9} = 4;
11-34-17-52-26-13-40-20-10-5-16-8-4-2-1: a(11) = #{3,6,7,9} = 4;
12-6-3-10-5-16-8-4-2-1: a(12) = #{7,9,11} = 3.

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

Cf. A159999, A220237, A070165.

import Data.List ((\\), genericIndex)
a246436 n = length $ [1..n] \\ genericIndex a220237_tabf (n - 1)

Table[Length[Complement[Range[n],NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&]]],{n,70}] (* Harvey P. Dale, May 27 2018 *)

A258824 Least number k such that A258822(k) = n.

Original entry on oeis.org

1, 2, 24, 63105

Offset: 0

Derek Orr, Jun 11 2015

If a(n) exists, a(n) > 10^6 for n > 3.
Excluding k = 24, for n = 2, after 29 and 34 iterations, you arrive at 29 and 34, respectively. Excluding k = 24, it appears all of the trajectories of the possible k values have length 48 or 49.
For n = 3, after 216, 234, and 252 iterations, you arrive at 216, 234, and 252, respectively. It appears all of the trajectories of the possible k values have length 317.

			For n = 24, the '3x+1' map is as follows: 24 -> 12 -> 6 -> 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1. After the 3rd iteration, we reach 3 and after the 5 iteration, we reach 5. Since 12 is the smallest number to have exactly two occurrences, a(2) = 24. Note that the length of this trajectory is 11. For all other trajectories with exactly two occurrences, the length is either 48 or 49.

Cf. A258822, A006370, A070165.

Tvect(n)=v=[n]; while(n!=1, if(n%2, k=3*n+1; v=concat(v, k); n=k); if(!(n%2), k=n/2; v=concat(v, k); n=k)); v
n=0; m=1; while(m<10^3, d=Tvect(m); c=0; for(i=1, #d, if(d[i]==i-1, c++)); if(c==n, print1(m, ", "); m=0; n++); m++)

A280409 Primes in the order that they appear in A280408, without repetitions.

Original entry on oeis.org

2, 3, 5, 7, 11, 17, 13, 23, 53, 19, 29, 41, 31, 47, 71, 107, 137, 103, 233, 263, 593, 167, 251, 283, 479, 719, 1619, 911, 1367, 577, 433, 61, 37, 59, 89, 67, 101, 43, 83, 73, 113, 79, 179, 269, 131, 197, 97, 149, 109, 173, 139, 157, 127, 191, 431, 647, 971

Offset: 1

Matthew Campbell, Jan 02 2017

nonn

David Radcliffe, Table of n, a(n) for n = 1..10000

Cf. A280408, A000040, A070165.

DeleteDuplicates@ Flatten@ Table[Select[NestWhileList[If[EvenQ@ #, #/2, 3 # + 1] &, n, # > 1 &], PrimeQ], {n, 200}] (* Michael De Vlieger, Jan 02 2017 *)

A301937 a(n) is the smallest number whose Collatz ('3x+1') trajectory crosses its initial value exactly n times.

Original entry on oeis.org

1, 3, 10, 7, 6, 9, 22, 19, 14, 25, 18, 83, 62, 33, 54, 559, 108, 109, 110, 97, 188, 147, 166, 221, 146, 171, 292, 129, 194, 257, 294, 313, 342, 399, 506, 609, 462, 353, 398, 531, 834, 471, 530, 1153, 9854, 417, 470, 627, 8758, 9853, 626, 9225, 18450, 20609, 23718

Offset: 0

Jon E. Schoenfield, May 05 2018

nonn

Records: 1, 3, 10, 22, 25, 83, 559, 609, 834, 1153, 9854, 18450, 20609, 23718, 31142, 35090, 41586, 80294, 283262, 377681, 427762, 789305, 887954, 887964, 1403202, 1752022, ..., . - Robert G. Wilson v, May 06 2018

			The Collatz trajectory for k=3 is 3 -> 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1 crosses the threshold value of 3 on exactly one iteration: the iteration on which it moves from 4 to 2. No smaller value of k shares this property, so a(1) = 3.
The Collatz trajectory for k=6 (see A304030) is nearly identical, containing, in order of appearance, the values 6, 3, 10, 5, 16, 8, 4, 2, 1; it crosses the threshold value of 6 on exactly 4 iterations (3 -> 10, 10 -> 5, 5 -> 16, and 8 -> 4). No smaller value of k shares this property, so a(4) = 6.

Jon E. Schoenfield, Table of n, a(n) for n = 0..185
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A070165, A304030.

nMax:=54; a:=[0: n in [1..nMax]]; for k in [2..24000] do n:=0; t:=k; while t gt 1 do tPrev:=t; if IsEven(t) then t:=t div 2; else t:=3*t+1; end if; if (t-k)*(tPrev-k) lt 0 then n+:=1; end if; end while; if (n gt 0) and (n le nMax) then if a[n] eq 0 then a[n]:=k; end if; end if; end for; a;

Collatz[n_] := NestWhileList[ If[ OddQ@#, 3# +1, #/2] &, n, # > 1 &]; f[n_] := Block[{x = Length[ SplitBy[ Collatz@ n, # < n +1 &]] - 1}, If[ OddQ@ n && n > 1, x - 1, x]]; t[] := 0; k = 1; While[k < 24000, If[ t[f[k]] == 0, t[f[k]] = k]; k++]; t@# & /@ Range@54 (* _Robert G. Wilson v, May 05 2018 *)

a(n) = min{k : A304030(k) = n}.

If the Collatz conjecture is true, then a(n) == n (mod 2) for all terms.

cp's OEIS Frontend

A221470 Least m such that the Collatz (3x+1) iteration of m has exactly n increasing peak values.

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A223699 Least number whose Collatz (3x+1) iteration has its maximum value at position n (counting from 1).

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Mathematica

A223700 The maximum value of the Collatz (3x+1) iteration beginning at A223699(n).

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Mathematica

A225570 The greedy smallest infinite reverse Collatz (3x+1) sequence.

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Mathematica

A225840 Largest number less than n occurring in Collatz trajectory starting with n.

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A226123 Number of terms of the form 2^k in Collatz(3x+1) trajectory of n.

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A246436 Number of numbers 1,...,n not occurring in the Collatz trajectory starting with n.

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A258824 Least number k such that A258822(k) = n.

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