A362703
Expansion of e.g.f. 1/(1 + LambertW(-x^3 * exp(x))).
Original entry on oeis.org
1, 0, 0, 6, 24, 60, 1560, 20370, 161616, 2601144, 53827920, 829605150, 14894289960, 360575394036, 8234733389064, 188800085076330, 5145737430116640, 148419618327231600, 4278452209330445856, 134018446273097264694, 4529883358179857555640
Offset: 0
A362604
Expansion of e.g.f. 1/(1 + LambertW(-x * exp(x^2))).
Original entry on oeis.org
1, 1, 4, 33, 352, 4805, 80256, 1582693, 36001792, 927974601, 26729943040, 850921057481, 29666297020416, 1124166449205709, 46005243970846720, 2022121401647311245, 95008417631810093056, 4751844218849365365137, 252063937292253895065600
Offset: 0
A367511
Highly composite numbers h(k) = A002182(k) such that h >= rad(h)^2, where rad() = A007947().
Original entry on oeis.org
1, 4, 36, 48, 45360, 50400
Offset: 1
Let P(n) = A002110(n).
a(1) = h(1) = 1 since 1 >= 1^2.
a(2) = h(3) = 4 since 4 >= P(1)^2, 4 >= 2^2.
a(3) = h(7) = 36 since 36 >= P(2)^2, 36 >= 6^2.
a(4) = h(8) = 48 since 48 >= P(2)^2, 48 >= 6^2.
a(5) = h(26) = 43560 since 43560 >= P(4)^2, where P(4) = 210, and 210^2 = 44100.
a(6) = h(27) = 50400 since 50400 >= P(4)^2.
Let V(i) = A301414(i) and let P(j) = A002110(j).
Plot of highly composite h = V(i)*P(j) at (x,y) = (j,i), i = 1..16, j = 1..7, showing h in this sequence in parentheses, and h in A168263 marked with an asterisk (*):
V(i)\P(j) 1 2 6 30 210 2310 30030 ...
+---------------------------------------
1 |(1*) 2* 6*
2 | (4*) 12* 60*
4 | 24* 120* 840*
6 | (36) 180* 1260*
8 | (48) 240 1680*
12 | 360 2520 27720*
24 | 720 5040 55440 720720
36 | 7560 83160 1081080
48 | 10080 110880 1441440
72 | 15120 166320 2162160
96 | 20160 221760 2882880
120 | 25200 277200 3603600
144 | 332640 4324320
216 | (45360) 498960 6486480
240 | (50400) 554400 7207200
...
Cf.
A001221,
A002110,
A002182,
A007947,
A025487,
A108602,
A126706,
A131605,
A168263,
A286708,
A301413,
A301414,
A303606,
A332785,
A365308,
A362702,
A366250.
-
(* First load function f at A025487, then run the following: *)
s = Union@ Flatten@ f[12];
t = Map[DivisorSigma[0, #] &, s];
h = Map[s[[FirstPosition[t, #][[1]]]] &, Union@ FoldList[Max, t]];
Reap[Do[If[# >= Product[Prime[j], {j, PrimeNu[#]}]^2, Sow[#]] &[ h[[i]] ],
{i, Length[h]}] ][[-1, 1]]
Showing 1-3 of 3 results.
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