A319865
Product of distinct prime factors of highly composite numbers (definition 1, A002182).
Original entry on oeis.org
1, 2, 2, 6, 6, 6, 6, 6, 30, 30, 30, 30, 30, 30, 210, 210, 210, 210, 210, 210, 210, 210, 210, 210, 2310, 210, 210, 2310, 2310, 2310, 2310, 2310, 2310, 2310, 2310, 2310, 2310, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 30030, 510510, 30030
Offset: 1
n | A002182(n) | a(n)
---+----------------------------------+--------------------------
24 | 25200 = 2^4 * 3^2 * 5^2 * 7 | 210 = 2 * 3 * 5 * 7
25 | 27720 = 2^3 * 3^2 * 5 * 7 * 11 | 2310 = 2 * 3 * 5 * 7 * 11
26 | 45360 = 2^4 * 3^4 * 5 * 7 | 210 = 2 * 3 * 5 * 7
27 | 50400 = 2^5 * 3^2 * 5^2 * 7 | 210 = 2 * 3 * 5 * 7
A365902
Irregular triangle of highly composite numbers h(n) = A002182(n) arranged first according to rad(h(n))/h(n) then by rad(h(n)), where rad(n) = A007947(n).
Original entry on oeis.org
1, 2, 6, 4, 12, 60, 24, 120, 840, 36, 180, 1260, 48, 240, 1680, 360, 2520, 27720, 720, 5040, 55440, 720720, 7560, 83160, 1081080, 10080, 110880, 1441440, 15120, 166320, 2162160, 36756720, 698377680, 20160, 221760, 2882880, 25200, 277200, 3603600, 61261200, 332640
Offset: 1
Row 1 contains the products of A301414(1) = 1 and each of P(0) = 1, P(1) = 2, and P(2) = 6.
Row 2 contains the products of A301414(2) = 2 and each of P(1), P(2), and P(3) = 30.
Row 3 contains the products of A301414(3) = 4 and each of P(2) and P(3), etc.
Table of first rows of S(n,j), where for S(n,j) = A002182(i), j = A108602(i):
n\j | 0 1 2 3 4 5
----------------------------------
1 | 1, 2, 6
2 | 4, 12, 60
3 | 24, 120
4 | 36, 180, 1260
5 | 48, 240, 1680
6 | 360, 2520, 27720
7 | 720, 5040, 720720, etc.
In this sequence T(n,k) we have the following:
1: 1, 2, 6;
2: 4, 12, 60;
3: 24, 120;
4: 36, 180, 1260;
5: 48, 240, 1680;
6: 360, 2520, 27720;
7: 720, 5040, 720720; etc.
-
nn = 8; rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
MapIndexed[Set[P[First[#2]], #1] &, FoldList[Times, Prime@ Range[nn + 1]]];
a2182 = Import["https://oeis.org/A002182/b002182.txt", "Data"][[All, -1]];
TakeWhile[
SplitBy[SortBy[
Map[{#1/#2, PrimeNu[#2], #1} & @@ {#, rad[#]} &,
TakeWhile[a2182, rad[#] <= P[nn] &]], #[[1 ;; 2]] &,
LexicographicOrder], First],
FreeQ[a2182, #1 P[#2 + 1]] & @@ #[[-1, 1 ;; 2]] &][[All, All, -1]] // Flatten
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]]
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