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A365900 Highly composite numbers k that remain highly composite when recursively divided by squarefree kernel.

%I A365900 #8 Feb 16 2025 08:34:06
%S A365900 1,2,4,6,12,24,36,60,120,180,360,720,840,1260,2520,5040,7560,25200,
%T A365900 27720,55440,83160,277200,720720,1081080,3603600,10810800,21621600,
%U A365900 61261200,183783600,367567200,3491888400,6983776800,48886437600,73329656400,80313433200,160626866400,1124388064800,1686582097200,32607253879200,48910880818800,1010824870255200,1516237305382800
%N A365900 Highly composite numbers k that remain highly composite when recursively divided by squarefree kernel.
%C A365900 Let h(n) = A002182(n).
%C A365900 Let f(x) = x/rad(x) = A301413(x), where rad(n) = A007947(n) is a primorial and x is in h.
%C A365900 If f(h(k)) = m is highly composite, then we apply f(m) until we reach 1 or m that is not highly composite.
%C A365900 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.
%C A365900 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.
%C A365900 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.
%H A365900 Michael De Vlieger, <a href="/A365900/a365900.png">Plot of h(k) = S(i)*P(omega(h(n))) at (x,y) = (i, omega(h(k))</a> highlighting k such that h(k) is in this sequence.
%H A365900 Achim Flammenkamp, <a href="http://wwwhomes.uni-bielefeld.de/achim/highly.html">Highly composite numbers</a>.
%H A365900 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HighlyCompositeNumber.html">Highly Composite Number</a>.
%e A365900 1 is in this sequence since f(1) = 1 and 1 is highly composite.
%e A365900 2 is in this sequence since f(2) = 1 and 1 is highly composite.
%e A365900 12 is in this sequence since f(12) = 2, and f(2) = 1, both highly composite.
%e A365900 48 is not in this sequence since f(48) = 48/6 = 8, and 8 is not highly composite.
%e A365900 Applying f recursively to h(128) = 1516237305382800 yields the following chain:
%e A365900 1516237305382800 -> 7560 -> 36 -> 6 -> 1, all highly composite. It seems that this is the largest term in the sequence.
%e A365900 .
%e A365900 Tree plot of terms:
%e A365900 1 --- 2 --- 4 --- 24 --- 720 --- 21621600
%e A365900    |     |     |      |       |- 367567200
%e A365900    |     |     |      |       |- 6983776800
%e A365900    |     |     |      |       |_ 160626866400
%e A365900    |     |     |      |
%e A365900    |     |     |      |- 5040 -- 48886437600
%e A365900    |     |     |      |       |- 1124388064800
%e A365900    |     |     |      |       |- 32607253879200
%e A365900    |     |     |      |       |_ 1010824870255200
%e A365900    |     |     |      |
%e A365900    |     |     |      |- 55440
%e A365900    |     |     |      |_ 720720
%e A365900    |     |     |
%e A365900    |     |     |- 120 -- 25200
%e A365900    |     |     |      |- 277200
%e A365900    |     |     |      |- 3603600
%e A365900    |     |     |      |_ 61261200
%e A365900    |     |     |
%e A365900    |     |     |_ 840
%e A365900    |     |
%e A365900    |     |-12 --- 360 -- 10810800
%e A365900    |     |     |      |- 183783600
%e A365900    |     |     |      |- 3491888400
%e A365900    |     |     |      |_ 80313433200
%e A365900    |     |     |
%e A365900    |     |     |- 2520
%e A365900    |     |     |_ 27720
%e A365900    |     |
%e A365900    |     |_60
%e A365900    |
%e A365900    |_ 6 -- 36 --- 7560 --- 73329656400
%e A365900          |     |        |- 1686582097200
%e A365900          |     |        |- 48910880818800
%e A365900          |     |        |_ 1516237305382800
%e A365900          |     |
%e A365900          |     |_ 83160 -- 1081080
%e A365900          |
%e A365900          |-180
%e A365900          |_1260
%t A365900 (* Program loads highly composite numbers from A002182 bfile *)
%t A365900 a2182 = Import["https://oeis.org/A002182/b002182.txt", "Data"][[All, -1]];
%t A365900 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]];
%t A365900 Select[Array[
%t A365900   NestWhileList[#/rad[#] &, a2182[[#]], And[# > 1, ! FreeQ[a2182, #]] &] &, 250],
%t A365900   Last[#] == 1 &][[All, 1]]
%Y A365900 Cf. A002110, A002182, A301413, A301414.
%K A365900 nonn,fini,full
%O A365900 1,2
%A A365900 _Michael De Vlieger_, Oct 06 2023