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

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

Offset: 1

Author

Michael De Vlieger, Oct 06 2023

Keywords

• nonn
• fini
• full

Comments

Let h(n) = A002182(n).

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.

Examples

			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
		

Links

• Michael De Vlieger, Plot of h(k) = S(i)*P(omega(h(n))) at (x,y) = (i, omega(h(k)) highlighting k such that h(k) is in this sequence.
• Achim Flammenkamp, Highly composite numbers.
• Eric Weisstein's World of Mathematics, Highly Composite Number.

Crossrefs

Cf. A002110, A002182, A301413, A301414.

Programs

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]]