

A365900


Highly composite numbers k that remain highly composite when recursively divided by squarefree kernel.


2



1, 2, 4, 6, 12, 24, 36, 60, 120, 180, 360, 720, 840, 1260, 2520, 5040, 7560, 25200, 27720, 55440, 83160, 277200, 720720, 1081080, 3603600, 10810800, 21621600, 61261200, 183783600, 367567200, 3491888400, 6983776800, 48886437600, 73329656400, 80313433200, 160626866400, 1124388064800, 1686582097200, 32607253879200, 48910880818800, 1010824870255200, 1516237305382800
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OFFSET

1,2


COMMENTS

Let f(x) = x/rad(x) = A301413(x), where rad(n) = A007947(n) is a primorial and x is in h.
If f(h(k)) = m is highly composite, then we apply f(m) until we reach 1 or m that is not highly composite.
Let S be the chain of highly composite terms that result from the recursion of f beginning with k in A002182. Terms in S are nondecreasing and each appear in this sequence. Example: beginning with h(51), we have {21621600, 720, 24, 4, 2, 1}. Terms that follow h(51) in the chain appear in this sequence.
There are 19 known terms j in S = A301414 = union({A301413}) that are highly composite. f(h(k)) = j is a necessary but insufficient condition for h(k) to appear in this sequence.
The numbers j in {48, 240, 10080, 15120, 20160, 50400, 17297280} do not yield terms in this sequence, because {48, 240, 10080, 50400} settle to 8, S(32) = h(22) = 15120 settles to 72, S(33) = h(23) = 20160 ends up at 96, and the largest of the 19 terms, S(62) = h(50) = 17297280 ends up at 576, all of which are not highly composite. It appears that there are only 19 terms that enable membership in this sequence.


LINKS



EXAMPLE

1 is in this sequence since f(1) = 1 and 1 is highly composite.
2 is in this sequence since f(2) = 1 and 1 is highly composite.
12 is in this sequence since f(12) = 2, and f(2) = 1, both highly composite.
48 is not in this sequence since f(48) = 48/6 = 8, and 8 is not highly composite.
Applying f recursively to h(128) = 1516237305382800 yields the following chain:
1516237305382800 > 7560 > 36 > 6 > 1, all highly composite. It seems that this is the largest term in the sequence.
.
Tree plot of terms:
1  2  4  24  720  21621600
     367567200
     6983776800
    _ 160626866400
   
    5040  48886437600
     1124388064800
     32607253879200
    _ 1010824870255200
   
    55440
   _ 720720
  
   120  25200
    277200
    3603600
   _ 61261200
  
  _ 840
 
 12  360  10810800
    183783600
    3491888400
   _ 80313433200
  
   2520
  _ 27720
 
 _60

_ 6  36  7560  73329656400
   1686582097200
   48910880818800
  _ 1516237305382800
 
 _ 83160  1081080

180
_1260


MATHEMATICA

(* Program loads highly composite numbers from A002182 bfile *)
a2182 = Import["https://oeis.org/A002182/b002182.txt", "Data"][[All, 1]];
rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
Select[Array[
NestWhileList[#/rad[#] &, a2182[[#]], And[# > 1, ! FreeQ[a2182, #]] &] &, 250],
Last[#] == 1 &][[All, 1]]


CROSSREFS



KEYWORD

nonn,fini,full


AUTHOR



STATUS

approved



