cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A072122 Numbers with 12 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

39, 78, 79, 153, 156, 157, 158, 305, 306, 307, 312, 314, 315, 316, 317, 610, 611, 612, 613, 614, 624, 628, 629, 630, 631, 632, 634, 647, 683, 687, 1220, 1221, 1222, 1224, 1226, 1228, 1229, 1241, 1248, 1256, 1257, 1258, 1260, 1261, 1262, 1264, 1265, 1268
Offset: 1

Views

Author

Jim Nastos, Jun 19 2002

Keywords

Comments

The Collatz (or 3x+1) function is f(x) = x/2 if x is even, 3x+1 if x is odd. The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached.

Examples

			trajectory: 39, 118, 59, 178, 89, 268, 134, 67, 202, 101, 304, 152, 76, 38, 19, 58, 29, 88, 44, 22, 11, 34, 17, 52, 26, 13, 40, 20, 10, 5, 16, 8, 4, 2, 1 has 12 odd numbers.

References

  • J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.

Links

Crossrefs

Cf. A062052-A062060.
Column k=12 of A354236.

Programs

A072466 Numbers with 11 odd integers in their Collatz (or 3x+1) trajectory.

Original entry on oeis.org

57, 59, 114, 115, 118, 119, 228, 229, 230, 236, 237, 238, 456, 458, 460, 461, 465, 472, 473, 474, 476, 477, 507, 513, 912, 916, 917, 920, 922, 930, 931, 943, 944, 945, 946, 947, 948, 949, 952, 954, 971, 987, 1014, 1015, 1025, 1026, 1027, 1031, 1129, 1131
Offset: 1

Views

Author

Jim Nastos, Jun 19 2002

Keywords

Comments

The Collatz (or 3x+1) function is f(x) = x/2 if x is even, 3x+1 if x is odd. The Collatz trajectory of n is obtained by applying f repeatedly to n until 1 is reached.

References

  • J. Shallit and D. Wilson, The "3x+1" Problem and Finite Automata, Bulletin of the EATCS #46 (1992) pp. 182-185.

Links

Crossrefs

Cf. A062052-A062060.
Column k=11 of A354236.

Programs

  • Maple
    b:= proc(n) option remember; irem(n, 2, 'r')+
      `if`(n=1, 0, b(`if`(n::odd, 3*n+1, r)))
    end:
q:= n-> is(b(n)=11):
select(q, [$1..2000])[];  # Alois P. Heinz, May 18 2022
  • Mathematica
    ocollQ[n_]:=Length[Select[NestWhileList[If[EvenQ[#],#/2,3#+1]&,n,#>1&],OddQ[#]&]]==11; Select[Range[1140],ocollQ[#]&] (* Jayanta Basu, May 28 2013 *)

A380244 The Collatz (or 3x+1) trajectory starting at a(n) contains exactly n odd integers and a(n) is the n-th number with this property.

Original entry on oeis.org

1, 10, 12, 68, 45, 30, 72, 101, 134, 179, 237, 314, 422, 551, 723, 509, 1282, 887, 1170, 1535, 2021, 1509, 1899, 2412, 1780, 2217, 3170, 3867, 2819, 3728, 2511, 3155, 3972, 2802, 3578, 2623, 3444, 4302, 3087, 3968, 2690, 1806, 2336, 1593, 2084, 2757, 1884, 2477
Offset: 1

Views

Author

Alois P. Heinz, Jan 17 2025

Keywords

Examples

			a(2) = 10 is the second integer (after 5) having exactly two odd integers in the Collatz trajectory: 10 -> 5 -> 16 -> 8 -> 4 -> 2 -> 1.

Links

Crossrefs

Main diagonal of A354236.
Cf. A006577, A006667, A078719, A337144.

Programs

  • Maple
    b:= proc(n) option remember; irem(n, 2, 'r')+
      `if`(n=1, 0, b(`if`(n::odd, 3*n+1, r)))
    end:
A:= proc() local h, p, q; p, q:= proc() [] end, 0;
      proc(n, k)
        if k=1 then return 2^(n-1) fi;
        while nops(p(k)) A(n$2):
seq(a(n), n=1..48);

Formula

A078719(a(n)) = n.

A375888 Rectangular array: row n shows all k such that n is the number of rises in the trajectory of k in the Collatz problem.

Original entry on oeis.org

1, 2, 5, 4, 10, 3, 8, 20, 6, 17, 16, 21, 12, 34, 11, 32, 40, 13, 35, 22, 7, 64, 42, 24, 68, 23, 14, 9, 128, 80, 26, 69, 44, 15, 18, 25, 256, 84, 48, 70, 45, 28, 19, 49, 33, 512, 85, 52, 75, 46, 29, 36, 50, 65, 43, 1024, 160, 53, 136, 88, 30, 37, 51, 66, 86, 57
Offset: 0

Views

Author

Clark Kimberling, Sep 11 2024

Keywords

Comments

Assuming that the Collatz conjecture (also known as the 3x+1 conjecture) is true, this is a permutation of the positive integers; viz., every positive integer occurs exactly once. Conjecture: every row contains a pair of consecutive integers.

Examples

			Corner:
   1     2     4     8    16    32    64   128   256   512  1024
   5    10    20    21    40    42    80    84    85   160   168
   3     6    12    13    24    26    48    52    53    96   104
  17    34    35    68    69    70    75   136   138   140   141
  11    22    23    44    45    46    88    90    92    93   176
   7    14    15    28    29    30    56    58    60    61   112
   9    18    19    36    37    38    72    74    76    77    81
6 is in row 2 because the trajectory, (6, 3, 10, 5, 16, 4, 2, 1), has exactly 2 rises: 3 to 10, and 5 to 16.

Links

Crossrefs

Cf. A000027, A000079 (row 1), A092893 (column 1), A006667, A070265, A078719.
Cf. A354236.

Programs

  • Mathematica
    t = Table[Count[Differences[NestWhileList[If[EvenQ[#], #/2, 3 # + 1] &, n, # > 1 &]], ? Positive], {n, 2048}]; (* after _Harvey P. Dale, A006667 *)
r[n_] := Flatten[Position[t, n - 1]];
Column[Table[r[n], {n, 1, 21}]] (* array *)
u = Table[r[k][[n + 1 - k]], {n, 1, 12}, {k, 1, n}]
Flatten[u] (* sequence *)

Formula

Transpose of the array in A354236.
Previous Showing 11-14 of 14 results.

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