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.
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
Keywords
Examples
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.
Programs
-
Mathematica
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 *) -
PARI
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
-
Sage
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
Formula
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
Extensions
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
Comments