A186109 -id:A186109 - cp's OEIS frontend

A186107 Numerator of the frequency of the n-th dropping time in the Collatz iteration.

1, 1, 1, 1, 3, 7, 3, 15, 85, 173, 119, 961, 663, 8045, 17637, 51033, 54475, 312455, 663535, 950235, 5936673, 1684037, 39993895, 87986917, 128989251, 205059181, 949737339, 2861515293, 400296173, 19018424205

Offset: 1

T. D. Noe, Feb 12 2011

The possible dropping times are in A020914. The denominators are in A186108. The cumulative frequency of the dropping time is A186109(n)/A186110(n).

			The frequencies are 1/2, 1/4, 1/16, 1/16, 3/128, 7/256, 3/256, 15/2048, 85/8192,....

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

Cf. A126241 (dropping times)

a(n) = numerator of A186009(n) / 2^A020914(n-1).

A186110 Denominator of the cumulative frequency of the dropping time in the Collatz iteration.

Original entry on oeis.org

2, 4, 16, 8, 128, 256, 16, 2048, 8192, 32768, 32768, 262144, 16384, 2097152, 8388608, 16777216, 33554432, 134217728, 536870912, 1073741824, 4294967296, 4294967296, 34359738368, 137438953472, 274877906944, 137438953472, 2199023255552, 4398046511104, 4398046511104, 35184372088832

Offset: 1

T. D. Noe, Feb 12 2011

The numerators are in A186109. The frequency of the n-th dropping time is A186107(n)/A186108(n).

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

Cf. A126241 (dropping times)

a(n) = denominator of sum_{k=1..n} A186009(k) / 2^A020914(k-1).

A182137 Size of the set of b for numbers of the form 2^n*x + b that cannot be the smallest element of a set giving a duration of infinite flight in the Collatz problem.

Original entry on oeis.org

1, 3, 6, 13, 28, 56, 115, 237, 474, 960, 1920, 3870, 7825, 15650, 31473, 63422, 126844, 254649, 509298, 1021248, 2050541, 4101082, 8219801, 16490635, 32981270, 66071490, 132455435, 264910870, 530485275, 1060970550, 2123841570, 4253619813, 8507239626, 17027951548, 34095896991, 68191793982, 136471574881, 272943149762, 546144278026, 1093108792776, 2186217585552

Offset: 1

Jérôme STORTI, Apr 14 2012

nonn

In the Collatz Problem A014682, it is possible to apply the algorithm to first degree polynomials like 2^n*x+b, where n is an integer and 0 <= b < 2^n. The iteration terminates by two cases:
1) a*x+b where a < 2^n: the polynomial is "minimized"
2) a*x+b where a is odd and a > 2^n, parity cannot be found. The polynomial cannot be minimized.
The sequence counts how many first degree polynomials end like first case for each n > 0.
The interest of this sequence is that every number that can be described by a minimized polynomial cannot be the smallest element of a set of value of T(n) = infinity.

			Example with 4x+b (0 <= b < 4):
4x is even, thus gives 2x, 2 < 4 (first case).
4x+1, is odd thus 3(4x+1)+1 = 12x+4 is even, thus (12x+4)/2/2=3x+1 3 < 4, first case.
4x+2 is even, (4x+2)/2=2x+1, 2 < 4, first case.
4x+3 with same way gives 9x+8. 9 is odd and 9 > 4, second case.
That explains why the second (n=2) term in sequence is 3.

Cf. A020914, A074473, A076227, A100982, A186109, A243115.

a[n_] := Module[{b, p0, p1, minimized = 0}, For[b = 1, b <= 2^n, b++, {p0, p1} = {b, 2^n}; While[Mod[p1, 2] == 0 && p1 >= 2^n, {p0, p1} = If[Mod[p0, 2] == 0, {p0/2, p1/2}, {3*p0+1, 3*p1}]; If[p1<2^n, minimized += 1]]]; minimized]; Table[Print[an = a[n]]; an, {n, 1, 40}] (* Jean-François Alcover, Feb 12 2014, translated from D. S. McNeil's Sage code *)

upto(P=18)= my(r=Vec([1, 1], P)); forstep(x=3,2^P,4, my(s=x, p=0); until(s<=x, s= if(s%2, 3*s+1, s)/2; if(p++ > P, next(2))); if((2^p>x), r[p]++)); for(i=2, #r, r[i]+= 2*r[i-1]); print(r); \\ Ruud H.G. van Tol, Mar 13 2023

def A182137(n):
    minimized = 0
    for b in range(2**n):
        p = [b, 2**n]
        while p[1] % 2 == 0 and p[1] >= 2**n:
            p[0],p[1] = [p[0]/2, p[1]/2] if p[0] % 2 == 0 else [3*p[0]+1, 3*p[1]]
        if p[1] < 2**n: minimized += 1
    return minimized # D. S. McNeil, Apr 14 2012

a(n) = 2^n - A076227(n) for n >= 2. - Ruud H.G. van Tol, Mar 13 2023

For n not in A020914, a(n) = 2*a(n-1). - Ruud H.G. van Tol, Apr 12 2023

More terms from D. S. McNeil, Apr 14 2012
a(31) from Jérôme STORTI, Apr 22 2012
a(32)-a(38) from Jérôme STORTI, Jul 21 2012
a(39) from Jérôme STORTI, Jul 26 2012
a(40) from Jérôme STORTI, Feb 08 2014
a(37) and a(39) corrected by Jérôme STORTI, Dec 29 2021

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A186107 Numerator of the frequency of the n-th dropping time in the Collatz iteration.

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A186110 Denominator of the cumulative frequency of the dropping time in the Collatz iteration.

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A182137 Size of the set of b for numbers of the form 2^n*x + b that cannot be the smallest element of a set giving a duration of infinite flight in the Collatz problem.

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