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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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).

Original entry on oeis.org

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

Views

Author

Amiram Eldar, Jul 06 2019

Keywords

Comments

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

Examples

			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)

Links

Crossrefs

Subsequence of A003586 and A309015.
Cf. A000005, A002201.

Programs

  • 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 *)

Extensions

More terms from Michael De Vlieger, Jul 11 2019

A355578 Numbers whose sum of 3-smooth divisors sets a new record.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 108, 144, 192, 216, 288, 324, 384, 432, 576, 648, 768, 864, 972, 1152, 1296, 1536, 1728, 1944, 2304, 2592, 2916, 3072, 3456, 3888, 4608, 5184, 5832, 6912, 7776, 8748, 9216, 10368, 11664, 13824, 15552, 17496
Offset: 1

Views

Author

Amiram Eldar, Jul 08 2022

Keywords

Comments

Numbers m such that A072079(m) > A072079(k) for all k < m.
All the terms are 3-smooth numbers (A003586).
Equivalently, 3-smooth numbers k such that A000203(k) sets a new record.
Analogous to highly abundant numbers (A002093) with 3-smooth numbers only.

Examples

			The numbers of 3-smooth divisors of the first 6 positive integers are 1, 3, 4, 7, 1 and 12. The record values, 1, 3, 4 and 12, occur at 1, 2, 3, 4 and 6, the first 5 terms of this sequence.

Links

Crossrefs

Subsequence of A003586.
A355579 is a subsequence.
Cf. A000203, A002093, A072079, A309015.

Programs

  • Mathematica
    s[n_] := Module[{e = IntegerExponent[n, {2, 3}], p}, p = {2, 3}^e; If[Times @@ p == n, (2^(e[[1]] + 1) - 1)*(3^(e[[2]] + 1) - 1)/2, 0]]; sm = 0; seq = {}; Do[sn = s[n]; If[sn > sm, sm = sn; AppendTo[seq, n]], {n, 1, 18000}]; seq
  • PARI
    lista(nmax) = {my(list = List(), smax = 0, e2, e3, s); for(n = 1, nmax, e2 = valuation(n, 2); e3 = valuation(n, 3); s = if(2^e2 * 3^e3 == n, (2^(e2 + 1) - 1)*(3^(e3 + 1) - 1)/2, 0); if(s > smax, smax = s;  listput(list, n))); Vec(list)};
  • Python
    from sympy import multiplicity as v
from itertools import count, takewhile
def f(n): return (2**(v(2, n)+1)-1) * (3**(v(3, n)+1)-1)//2
def smooth3(lim):
    pows2 = list(takewhile(lambda x: x record: record = v; records.append(argv)
    return records
print(aupto(10**5)) # Michael S. Branicky, Jul 08 2022
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