A126241 -id:A126241 - cp's OEIS frontend

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

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

A381707 Smallest initial value for unimodal Collatz (3x+1)/2 glide sequence that begins with exactly n increases.

Original entry on oeis.org

5, 3, 23, 15, 95, 575, 383, 255, 5631, 25599, 104447, 69631, 745471, 3293183, 2195455, 12648447, 97910783, 65273855, 43515903, 1460666367, 6700400639, 4466933759, 71697432575, 47798288383, 764873277439, 1242923270143, 3760646520831, 8371159695359, 5580773130239, 3720515420159

Offset: 1

David Dewan, Mar 04 2025

nonn

A unimodal Collatz glide sequence is successive rises x -> (3x+1)/2 followed by successive falls x -> x/2 until dropping below its starting x.

After n increases, there are ceiling(n*log(3)/log(2) - n) decreases to drop below the initial value.

			For n=3, the smallest starting x = a(3) = 23 has trajectory
  23 - 35 -> 53 ->  80  -> 40 -> 20
     \-----------/      \------/
     n=3 increases   decreases to < initial

David Dewan, Unimodal Glide Sequences

Cf. A126241, A122437, A122458.

a[n_]:=2^n ModularInverse[3^n,2^Max[Ceiling[Log2[3^n]-n],2]]-1; Array[a,30]

a(n)={my(m=2^(logint(3^n,2) - n + 1 + (n==1))); 2^n*lift(1/Mod(3^n,m)) - 1} \\ Andrew Howroyd, Mar 09 2025

a(n) = 2^n * (3^(-n) mod 2^max(2, ceiling(log2(3^n)-n))) - 1.

A198724 Let P(n) be the maximal prime divisor of 3*n+1. Then a(n) is the smallest number of iterations of P(n) such that the a(n)-th iteration < n, and a(n) = 0, if such number does not exist.

Original entry on oeis.org

2, 3, 1, 6, 4, 1, 1, 6, 3, 2, 1, 2, 2, 1, 1, 1, 2, 3, 1, 3, 1, 2, 1, 2, 6, 1, 1, 1, 4, 3, 1, 2, 2, 3, 1, 1, 5, 1, 1, 3, 1, 1, 1, 2, 3, 1, 1, 3, 1, 6, 1, 2, 2, 1, 1, 1, 4, 2, 1, 1, 1, 2, 1, 2, 2, 1, 1, 3, 2, 1, 1, 2, 2, 2, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 3, 1

Offset: 3

Vladimir Shevelev, Oct 29 2011

nonn

Question. Is the sequence bounded?

By private communication from Alois P. Heinz, the places of records are 3, 4, 6, 286, 29866 with values 2, 3, 6, 8, 10. No more up to 46000000.

			For n=52 we have iterations: P^(1)=157, P^(2)=59, P^(3)=89, P^(4)=67, P^(5)=101, P^(6)=19<52. Thus a(52)=6.

V. Shevelev, Collatz-like problem with prime iterations

Cf. A074473, A126241.

P[n_] := FactorInteger[3*n + 1][[-1, 1]]; Table[k = 1; m = n; While[m = P[m]; m >= n, k++]; k, {n, 3, 100}] (* T. D. Noe, Oct 30 2011 *)

a(n) = {nb = 1; na = n; while((nna=vecmax(factor(3*na+1)[,1])) >= n,na = nna; nb++); nb;} \\ Michel Marcus, Feb 06 2016

A318759 Numbers x whose trajectory reaches 1 under recursive applications of the map x -> x/3 if x == 0 (mod 3), x -> (4x+2)/3 if x == 1 (mod 3), x -> (4x+1)/3 if x == 2 (mod 3).

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 13, 18, 20, 27, 29, 36, 39, 40, 54, 60, 65, 81, 87, 108, 109, 117, 120, 121, 136, 146, 162, 180, 182, 195, 197, 243, 245, 261, 263, 272, 324, 327, 328, 332, 351, 360, 363

Offset: 1

Jack Warren, Sep 02 2018

nonn

Cf. A014682, A126241, A138754.

cp's OEIS Frontend

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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A381707 Smallest initial value for unimodal Collatz (3x+1)/2 glide sequence that begins with exactly n increases.

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A198724 Let P(n) be the maximal prime divisor of 3*n+1. Then a(n) is the smallest number of iterations of P(n) such that the a(n)-th iteration < n, and a(n) = 0, if such number does not exist.

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A318759 Numbers x whose trajectory reaches 1 under recursive applications of the map x -> x/3 if x == 0 (mod 3), x -> (4x+2)/3 if x == 1 (mod 3), x -> (4x+1)/3 if x == 2 (mod 3).

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cp's OEIS Frontend

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

Views

Author

Keywords

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Examples

Crossrefs

Programs

PARI

Extensions

A381707 Smallest initial value for unimodal Collatz (3x+1)/2 glide sequence that begins with exactly n increases.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A198724 Let P(n) be the maximal prime divisor of 3*n+1. Then a(n) is the smallest number of iterations of P(n) such that the a(n)-th iteration < n, and a(n) = 0, if such number does not exist.

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Mathematica

PARI

A318759 Numbers x whose trajectory reaches 1 under recursive applications of the map x -> x/3 if x == 0 (mod 3), x -> (4*x+2)/3 if x == 1 (mod 3), x -> (4*x+1)/3 if x == 2 (mod 3).

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A318759 Numbers x whose trajectory reaches 1 under recursive applications of the map x -> x/3 if x == 0 (mod 3), x -> (4x+2)/3 if x == 1 (mod 3), x -> (4x+1)/3 if x == 2 (mod 3).