|
| |
|
|
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).
|
|
2
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
OFFSET
|
1,2
|
|
|
COMMENTS
|
|
|
|
LINKS
|
|
|
|
EXAMPLE
|
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
|
|
|
|
KEYWORD
|
nonn
|
|
|
AUTHOR
|
|
|
|
EXTENSIONS
|
|
|
|
STATUS
|
approved
|
| |
|
|
