A360589 Numbers k that set records in A355432.

1, 18, 48, 54, 162, 384, 486, 1350, 1458, 2250, 2430, 3750, 6000, 6750, 7290, 11250, 12150, 14580, 15000, 15360, 18750, 21870, 30720, 33750, 36450, 37500, 43740, 56250, 61440, 65610, 93750, 122880, 168750, 182250, 187500, 196830, 245760, 281250, 328050, 360150, 375000, 393660

Offset: 1

Author

Michael De Vlieger, Feb 22 2023

Keywords

• nonn

Comments

Subset of A055932.

For n > 1, subset of A360768, which is in turn a subset of A126706.

Conjecture: for n > 2, subset of A364702. - Michael De Vlieger, Oct 04 2024

Examples

			Let rad(m) = A007947(m).
a(1) = 1 since 1 is the empty product.
a(2) = 18 since {12} is a nondivisor k < 18 such that rad(k) = rad(18).
a(3) = 48 since {18, 36} are nondivisors k < 48 such that rad(k) = rad(48).
a(4) = 54 since {12, 24, 36, 48} are nondivisors k < 54 such that rad(k) = rad(54), etc.
Table shows prime decomposition of a(n) = Product p^e, noting multiplicity e in the pi(p)-th position. For example, a(n) = 1350 = 2 * 3^3 * 5^2, hence we write 1.3.2.
a(n) = A055932(i) and has A360912(n) nondivisors k < a(n) such that rad(k) = rad(a(n)).
   n    a(n) A067255(a(n))  i  A360912(n)
  ----------------------------------------
   1      1      0          1          0
   2     18      1.2        8          1
   3     48      4.1       13          2
   4     54      1.3       14          4
   5    162      1.4       25          8
   6    384      7.1       37         10
   7    486      1.5       42         14
   8   1350      1.3.2     65         16
   9   1458      1.6       67         21
  10   2250      1.2.3     81         23
  11   2430      1.5.1     85         26
  12   3750      1.1.4     99         33
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..2071
• Michael De Vlieger, Plot p^m | a(n) at (x,y) = (n, pi(p)), n = 1..2071, 24X vertical exaggeration, with a color function that represents m = 1 in black, m = 2 in red, m = 3 in orange, ... m = 34 in magenta. (Represents column "A067255(a(n))" in table in Example below.)

Crossrefs

Cf. A007947, A055932, A126706, A355432, A360768, A360912, A364702.

Programs

Mathematica

rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]]; t = Select[Range[2^14], Nor[SquareFreeQ[#], PrimePowerQ[#]] &]; s = Select[t, #1/#2 >= #3 & @@ {#1, Times @@ #2, #2[[2]]} & @@ {#, FactorInteger[#][[All, 1]]} &]; t = Table[m = s[[n]]; r = rad[m]; Count[TakeWhile[t, # < m &], _?(And[rad[#] == r, Mod[m, #] != 0] &)], {n, Length[s]}]; {1}~Join~Map[s[[FirstPosition[t, #][[1]]]] &, Union@ FoldList[Max, t]]