A166245 -id:A166245 - cp's OEIS frontend

A260590 The modified Syracuse algorithm, msa, applied to 2n+1.

4, 2, 7, 2, 5, 2, 7, 2, 4, 2, 5, 2, 59, 2, 56, 2, 4, 2, 8, 2, 5, 2, 54, 2, 4, 2, 5, 2, 7, 2, 54, 2, 4, 2, 51, 2, 5, 2, 8, 2, 4, 2, 5, 2, 45, 2, 8, 2, 4, 2, 42, 2, 5, 2, 31, 2, 4, 2, 5, 2, 8, 2, 15, 2, 4, 2, 7, 2, 5, 2, 7, 2, 4, 2, 5, 2, 40, 2, 21, 2, 4, 2, 29, 2, 5, 2, 8, 2, 4, 2, 5, 2, 7, 2, 13

Offset: 1

Joseph K. Horn and Robert G. Wilson v, Jul 29 2015

nonn

Normally the '3x+1 problem' or 'Collatz problem' asks for the number of steps to go from n to 1 (A006577). Here we ask for the number of iterations of the mapping, msa, to go from n to less than n; the mapping of x is either -> (3x+1)/2 if x is odd or -> x/2 if x is even.
Since the number of iterations of msa for an even number is always 1, we will only investigate the odd numbers greater than one.
a(n) = 1 for no values of n;
a(n) = 2 for n = 2 + 2k (k=0,1,2,3,...);
a(n) = 3 for no values of n;
a(n) = 4 for n = 1 + 8k (k=0,1,2,3,...);
a(n) = 5 for n = 5 + 16k and 11 + 16k (k=0,1,2,3,...);
a(n) = 6 for no values of n;
a(n) = 7 for n = 3 + 64k, 7 + 64k, 29 + 64k, etc. (k=0,1,2,3,...).
Possible values for a(n) are: 2, 4, 5, 7, 8, 10, 12, 13, 15, 16, 18, 20, 21, 23, 24, 26, 27, 29, ... (A260593, sorted). Density is ~ 5/8.
Record values: 4, 7, 59, 81, 105, 135, 164, 165, 173, 176, 183, 224, 246, 287, 292, 298, 308, 376, 395, 398, 433, 447, 547, ....
And the records occur for n: 1, 3, 13, 351, 5043, 17827, 135135, 181171, 190863, 313165, 513715, 563007, 4044031, 6710835, 10319167, 13358335, 28462477, 31864063, 108870007, 600495895, 913698783, 1394004493, ....
Remember these n-values are the indices of odd numbers (A005408).

			a(1) is 4 because 2n+1 is 3 and 3 -> 5 -> 8 -> 4 -> 2. The number of iterations of the msa is 4;
a(2) is 2 because 2n+1 is 5 and 5 -> 8 -> 4. The number of iterations of the msa is 2;
a(3) is 7 because 2n+2 is 7 and 7 -> 11 -> 17 -> 26 -> 13 -> 20 -> 10 -> 5. The number of iterations of the msa is 7; etc.
Also see The Modified Syracuse Algorithm link.

Danny Rorabaugh, Table of n, a(n) for n = 1..10000
Encyclopedia of Mathematics, Syracuse problem.
Joseph K. Horn, HHC 2014, HP Handheld Conference, Sept. 20-21, 2014, Reno, NV, Hailstone Numbers: A Pattern Has Been Found.
Joseph K. Horn, The Modified Syracuse Algorithm.
Eric Weisstein's World of Mathematics, The Syracuse Algorithm
Wikipedia, Collatz conjecture. particularly Section "Cycles".

Cf. A005408, A006577, A020857, A075677, A075884, A076536, A144396, A166245.

msa[n_] := If[ OddQ@ n, (3n + 1)/2, n/2]; f[n_] := Block[{k = 2n + 1}, Length@ NestWhileList[ msa@# &, k, # >= k &] - 1]; Array[f, 95]

a(n) = the number of iterations for the msa; i.e., the number of mappings of x -> (3x+1)/2 if x is odd or -> x/2 if x is even to arrive at a number less than n.

a(n) = the binary length of A260592(n).

A365478 In the Collatz problem, largest value in the trajectory of n in the 3x+1 function (denoted by T(x) in the literature, and defined as T(x) = (3x+1)/2 if x is odd, T(x) = x/2 if x is even), or -1 if the trajectory is divergent.

Original entry on oeis.org

1, 2, 8, 4, 8, 8, 26, 8, 26, 10, 26, 12, 20, 26, 80, 16, 26, 26, 44, 20, 32, 26, 80, 24, 44, 26, 4616, 28, 44, 80, 4616, 32, 50, 34, 80, 36, 56, 44, 152, 40, 4616, 42, 98, 44, 68, 80, 4616, 48, 74, 50, 116, 52, 80, 4616, 4616, 56, 98, 58, 152, 80, 92, 4616, 4616

Offset: 1

Paolo Xausa, Sep 05 2023

nonn

This sequence differs from A025586, where the division by 2 does not immediately follow the 3x+1 step when x is odd.
Here by definition the trajectory ends when 1 is reached, so a(1) = 1.
Kontorovich and Lagarias (2009, 2010) call these values the maximum excursion values.

			a(11) = 26 because 26 is the largest value in the trajectory 11 -> 17 -> 26 -> 13 -> 20 -> 10 -> 5 -> 8 -> 4 -> 2 -> 1.

Paolo Xausa, Table of n, a(n) for n = 1..10000
Alex V. Kontorovich and Jeffrey C. Lagarias, Stochastic Models for the 3x+1 and 5x+1 Problems, arXiv:0910.1944 [math.NT], 2009, pp. 11-14, and in Jeffrey C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010, pp. 140-142.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A014682, A025586 (equivalent for the Collatz function), A166245.

A365478[n_]:=Max[NestWhileList[If[OddQ[#],(3#+1)/2,#/2]&,n,#>1&]];Array[A365478,100]

a(n) <= A025586(n).

A349325 Number of times the Collatz plot started at n crosses the y = n line, or -1 if the number of crossings is infinite.

Original entry on oeis.org

1, 1, 2, 1, 2, 3, 4, 1, 4, 1, 2, 1, 2, 5, 2, 1, 4, 5, 6, 1, 2, 3, 2, 1, 6, 1, 4, 1, 2, 3, 4, 1, 6, 1, 2, 1, 2, 3, 4, 1, 6, 1, 4, 1, 2, 3, 4, 1, 4, 1, 2, 1, 2, 7, 6, 1, 6, 1, 2, 3, 4, 7, 6, 1, 4, 1, 2, 1, 2, 3, 6, 1, 10, 1, 2, 1, 2, 5, 2, 1, 4, 5, 6, 1, 2, 3, 2

Offset: 1

Paolo Xausa, Nov 15 2021

nonn

The plots considered are the trajectories from n to 1 of the 3x+1 function, denoted by T(x) in the literature, defined as T(x) = (3x+1)/2 if x is odd, T(x) = x/2 if x is even (A014682).
The starting value of the trajectory is considered a crossing.
A similar sequence for the "standard" Collatz function (A006370) is A304030.
Conjecture: every positive integer appears in the sequence infinitely many times.

			The trajectory starting at 7 is 7 -> 11 -> 17 -> 26 -> 13 -> 20 -> 10 -> 5 -> 8 -> 4 -> 2 -> 1, so the Collatz plot crosses the y = 7 line at the beginning, from 10 to 5, from 5 to 8 and from 8 to 4, for a total of 4 times. a(7) is therefore 4.

J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, American Mathematical Society, 2010.

J. C. Lagarias, The 3x+1 Problem: An Overview, arXiv:2111.02635 [math.NT], 2021.
Index entries for sequences related to 3x+1 (or Collatz) problem

Cf. A006370, A014682, A070168, A166245, A304030.

nterms=100;Table[h=1;prevc=c=n;While[c>1,If[OddQ[c],c=(3c+1)/2;If[prevcn,h++],c/=2^IntegerExponent[c,2];If[prevc>n&&c

f(n) = if (n%2, (3*n+1)/2, n/2); \\ A014682
a(n) = {my(nb=1, prec=n, next); while (prec != 1, next = f(prec); if ((next-n)*(prec-n) <0, nb++); prec = next;); nb;} \\ Michel Marcus, Nov 16 2021

def A349325(n):
    prevc = c = n
    h = 1
    while c > 1:
        if c % 2:
            c = (3*c+1) // 2
            if prevc < n and c > n: h += 1
        else:
            c //= 2
            if prevc > n and c < n: h += 1
        prevc = c
    return h
print([A349325(n) for n in range(1, 100)])

a(2^k) = 1, for integers k >= 0.

a(A166245(m)) = 1 for m>=1. - Michel Marcus, Nov 16 2021

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).

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A260590 The modified Syracuse algorithm, msa, applied to 2n+1.

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A365478 In the Collatz problem, largest value in the trajectory of n in the 3x+1 function (denoted by T(x) in the literature, and defined as T(x) = (3x+1)/2 if x is odd, T(x) = x/2 if x is even), or -1 if the trajectory is divergent.

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A349325 Number of times the Collatz plot started at n crosses the y = n line, or -1 if the number of crossings is infinite.

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

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