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

Offset

1,2

Comments

Subset of A055932.

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

Links

Michael De Vlieger, Table of n, a(n) for n = 1..2071

Example

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
  ...

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]]

Crossrefs

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

Sequence in context: A034132 A218630 A099119 * A105520 A067726 A234956

Adjacent sequences: A360586 A360587 A360588 * A360590 A360591 A360592

Keyword

nonn

Author

Michael De Vlieger, Feb 22 2023

Status

approved