A339769
Number of unique heights that are achieved by only one starting number in the Collatz (or '3x+1') problem when starting from numbers in the range [2^n, 2^(n+1)).
Original entry on oeis.org
1, 2, 4, 4, 5, 6, 10, 12, 9, 7, 4, 9, 13, 11, 7, 6, 8, 10, 13, 14, 10, 15, 14, 21, 8, 7, 13, 21
Offset: 0
a(5)=6 since the 6 heights 5, 26, 34, 109, 29, 104 are uniquely attained from the starting numbers 32, 33, 39, 41, 43, 47, respectively. The largest of the distinct heights (A280341) in the interval [32,64) however is 112.
a(11)=9 with largest unique height 237 for starting value 3711 in interval [2^11, 2^12) also is the largest height for all starting values in the interval.
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collatz[n_] := If[EvenQ[n], n/2, 3n+1]
height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1
a339769[n_] := Module[{heightL={}, countL={}, s, h, p}, For[s=2^n, s<2^(n+1), s++, h=height[s]; If[!MemberQ[heightL, h], AppendTo[heightL, h]; AppendTo[countL, 1], {{p}}=Position[heightL, h]; countL[[p]]+=1]]; Length[Select[Transpose[{heightL, countL}], #[[2]]==1&]]]
(* sequence data; long computation times for n >= 22 *)
Map[a339769, Range[0, 27]]
A335569
a(n) is the maximum height achieved in the Collatz ('3x+1') problem when starting from numbers in the range [2^n, 2^(n+1)).
Original entry on oeis.org
0, 7, 16, 19, 111, 112, 118, 127, 143, 178, 181, 237, 261, 275, 307, 339, 353, 442, 469, 524, 556, 596, 664, 704, 705, 949, 950, 956, 964, 986, 1008, 1050, 1131, 1210, 1219, 1220, 1234, 1307, 1321
Offset: 0
a(35) = 1220 is the smallest term having 3 start values achieving maximum height: 63389366646, 63389366647, 64375365601. - _Bert Dobbelaere_, Feb 13 2021
-
collatz[n_] := If[EvenQ[n], n/2, 3n+1]
height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1
a335569[n_] := Max[Map[height, Range[2^n, 2^(n+1)-1]]]
(* sequence data; long computation times for n >= 22 *)
Map[a335569, Range[0, 27]]
A339773
a(n) is the first number k such that the n Collatz runs starting at the consecutive numbers k, k+1, ..., k+n-1 all have the same prime-valued height while the runs starting at k-1 and k+n have nonprime heights.
Original entry on oeis.org
25, 14, 108, 314, 1154, 840, 3360, 1494, 24408, 4722, 6576, 33578, 124097, 61442, 99248, 104879, 228296, 302956, 203436, 269698, 106122, 470826, 614402, 701224, 589826, 1369884, 252548, 1377184, 3126172, 1356161, 1370050, 1591584, 2065786, 8363804, 2054827
Offset: 1
a(2) = 14 since the 2 adjacent numbers 14 and 15 are the first consecutive 2 whose height in their Collatz runs is the same prime number, in this case 17, while the heights for the Collatz runs at 13 and 16 are the nonprimes 9 and 4, respectively.
a(9) = 24408 since the 9 adjacent numbers 24408 .. 24416 are the first consecutive 9 whose height in their Collatz runs is the same prime number, in this case 157, while the heights for the Collatz runs at 24407 and 24417 are the nonprimes 64 and 69, respectively.
-
collatz[n_] := If[EvenQ[n], n/2, 3n+1]
height[n_] := Length[NestWhileList[collatz, n, #!=1&]] - 1
(* b is an estimate on the size of the list being computed *)
a339773[n_, b_] := Module[{k=2, c, d, j, pList=Table[0, {b}]}, While[k<=n, c=height[k-1]; d=height[k]; j=k+1; If[!PrimeQ[c]&&PrimeQ[d], While[height[j]==d, j++]; If[!PrimeQ[height[j]]&&pList[[j-k]]==0, pList[[j-k]]=k]]; k=j]; pList]
Take[a339773[5000000, 50], 33] (* sequence data *)
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