A309016 Superior 2-highly composite numbers: 3-smooth numbers (A003586) k for which there is a real number e > 0 such that d(k)/k^e >= d(j)/j^e for all 3-smooth numbers j, where d(k) is the number of divisors of k (A000005).

1, 2, 6, 12, 24, 72, 144, 288, 864, 1728, 5184, 10368, 20736, 62208, 124416, 373248, 746496, 1492992, 4478976, 8957952, 26873856, 53747712, 107495424, 322486272, 644972544, 1289945088, 3869835264, 7739670528, 23219011584, 46438023168, 92876046336, 278628139008, 557256278016

Offset

1,2

Comments

How is this related to A163895? - R. J. Mathar, May 05 2023

Links

Michael De Vlieger, Table of n, a(n) for n = 1..2709

Gérard Bessi, Etude des nombres 2-hautement composés, Séminaire de Théorie des nombres de Bordeaux, Vol. 4 (1975), pp. 1-22.

Michael De Vlieger, Factors p analogous to A000705 such that the product of the smallest n terms equals a(n + 1) (10^5 terms).

Example

From Michael De Vlieger, Jul 12 2019: (Start)
We can plot all terms in A003586 with the power range 2^x with x >= 0 and 3^y with y >= 0 on the x and y axis, respectively. Plot of terms m in A309015, with terms also in a(n) placed in brackets:
                                2^x
          0    1     2     3     4     5     6     7     8
        +-----------------------------------------------------
     0  |[1]  [2]    4
     1  |     [6]  [12]  [24]   48
3^y  2  |           36   [72] [144]  [288]   576
     3  |                216   432   [864] [1728] 3456  6912 ...
          ...
Larger scale plot with "." representing a term m in A309015, and "o" representing a term in A309015 also in a(n) for all m < A002110(20).
                              2^x
        0    5   10   15   20   25   30   35   40   45  ...
        +------------------------------------------------
       0|oo.
        | ooo.
        |  .ooo.
        |   ..oo..
        |    ..ooo..
       5|      ..oo...
        |       ..ooo...
        |         ..oo....
        |          ..ooo....
        |            ..ooo....
      10|             ...oo.....
        |               ..ooo....
        |                ...oo.....
        |                  ..ooo.....
3^y     |                   ...ooo....
      15|                     ...oo.....
        |                      ...ooo.....
        |                        ...oo.....
        |                         ...ooo.....
        |                           ...oo......
      20|                            ...ooo.....
        |                              ...ooo.....
        |                               ....oo......
        |                                 ...ooo.....
        |                                  ....oo......
      25|                                    ...ooo......
        |                                     ....ooo....
        |                                       ....oo.
        |                                        ....o
        |                                          .
     ...
(End)

Mathematica

f[nn_, k_: 2] := Block[{w = {{2, 1}, {3, 0}}, s = {2}, P = 1, q = k - 2, x, i, n, f}, f[w_List] := Log[#1, (#2 + 2)/(#2 + 1)] & @@ w; x = Array[f[w[[#]] ] &, P + 1]; For[n = 2, n <= nn, n++, i = First@ FirstPosition[x, Max[x]]; AppendTo[s, w[[i, 1]]]; w[[i, 2]]++; If[And[i > P, P <= q], P++; AppendTo[w, {Prime[i + 1], 0}]; AppendTo[x, f[Last@ w]]]; x[[i]] = f@ w[[i]] ]; s]; {1}~Join~FoldList[Times, f[32, 2]] (* Michael De Vlieger, Jul 11 2019, after T. D. Noe at A000705 *)

Crossrefs

Subsequence of A003586 and A309015.

Cf. A000005, A002201.

Sequence in context: A163264 A335327 A163895 * A302652 A180071 A034882

Adjacent sequences: A309013 A309014 A309015 * A309017 A309018 A309019

Keyword

nonn

Author

Amiram Eldar, Jul 06 2019

Extensions

More terms from Michael De Vlieger, Jul 11 2019

Status

approved