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%I #15 May 05 2023 13:06:55
%S 1,2,6,12,24,72,144,288,864,1728,5184,10368,20736,62208,124416,373248,
%T 746496,1492992,4478976,8957952,26873856,53747712,107495424,322486272,
%U 644972544,1289945088,3869835264,7739670528,23219011584,46438023168,92876046336,278628139008,557256278016
%N 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).
%C How is this related to A163895? - _R. J. Mathar_, May 05 2023
%H Michael De Vlieger, <a href="/A309016/b309016.txt">Table of n, a(n) for n = 1..2709</a>
%H Gérard Bessi, <a href="https://eudml.org/doc/181966">Etude des nombres 2-hautement composés</a>, Séminaire de Théorie des nombres de Bordeaux, Vol. 4 (1975), pp. 1-22.
%H Michael De Vlieger, <a href="/A309016/a309016.txt">Factors p analogous to A000705 such that the product of the smallest n terms equals a(n + 1)</a> (10^5 terms).
%e From _Michael De Vlieger_, Jul 12 2019: (Start)
%e 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:
%e 2^x
%e 0 1 2 3 4 5 6 7 8
%e +-----------------------------------------------------
%e 0 |[1] [2] 4
%e 1 | [6] [12] [24] 48
%e 3^y 2 | 36 [72] [144] [288] 576
%e 3 | 216 432 [864] [1728] 3456 6912 ...
%e ...
%e 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).
%e 2^x
%e 0 5 10 15 20 25 30 35 40 45 ...
%e +------------------------------------------------
%e 0|oo.
%e | ooo.
%e | .ooo.
%e | ..oo..
%e | ..ooo..
%e 5| ..oo...
%e | ..ooo...
%e | ..oo....
%e | ..ooo....
%e | ..ooo....
%e 10| ...oo.....
%e | ..ooo....
%e | ...oo.....
%e | ..ooo.....
%e 3^y | ...ooo....
%e 15| ...oo.....
%e | ...ooo.....
%e | ...oo.....
%e | ...ooo.....
%e | ...oo......
%e 20| ...ooo.....
%e | ...ooo.....
%e | ....oo......
%e | ...ooo.....
%e | ....oo......
%e 25| ...ooo......
%e | ....ooo....
%e | ....oo.
%e | ....o
%e | .
%e ...
%e (End)
%t 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 *)
%Y Subsequence of A003586 and A309015.
%Y Cf. A000005, A002201.
%K nonn
%O 1,2
%A _Amiram Eldar_, Jul 06 2019
%E More terms from _Michael De Vlieger_, Jul 11 2019
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