This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A360589 #15 Oct 04 2024 16:52:37
%S A360589 1,18,48,54,162,384,486,1350,1458,2250,2430,3750,6000,6750,7290,11250,
%T A360589 12150,14580,15000,15360,18750,21870,30720,33750,36450,37500,43740,
%U A360589 56250,61440,65610,93750,122880,168750,182250,187500,196830,245760,281250,328050,360150,375000,393660
%N A360589 Numbers k that set records in A355432.
%C A360589 Subset of A055932.
%C A360589 For n > 1, subset of A360768, which is in turn a subset of A126706.
%C A360589 Conjecture: for n > 2, subset of A364702. - _Michael De Vlieger_, Oct 04 2024
%H A360589 Michael De Vlieger, <a href="/A360589/b360589.txt">Table of n, a(n) for n = 1..2071</a>
%H A360589 Michael De Vlieger, <a href="/A360589/a360589.png">Plot p^m | a(n) at (x,y) = (n, pi(p))</a>, 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.)
%e A360589 Let rad(m) = A007947(m).
%e A360589 a(1) = 1 since 1 is the empty product.
%e A360589 a(2) = 18 since {12} is a nondivisor k < 18 such that rad(k) = rad(18).
%e A360589 a(3) = 48 since {18, 36} are nondivisors k < 48 such that rad(k) = rad(48).
%e A360589 a(4) = 54 since {12, 24, 36, 48} are nondivisors k < 54 such that rad(k) = rad(54), etc.
%e A360589 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.
%e A360589 a(n) = A055932(i) and has A360912(n) nondivisors k < a(n) such that rad(k) = rad(a(n)).
%e A360589 n a(n) A067255(a(n)) i A360912(n)
%e A360589 ----------------------------------------
%e A360589 1 1 0 1 0
%e A360589 2 18 1.2 8 1
%e A360589 3 48 4.1 13 2
%e A360589 4 54 1.3 14 4
%e A360589 5 162 1.4 25 8
%e A360589 6 384 7.1 37 10
%e A360589 7 486 1.5 42 14
%e A360589 8 1350 1.3.2 65 16
%e A360589 9 1458 1.6 67 21
%e A360589 10 2250 1.2.3 81 23
%e A360589 11 2430 1.5.1 85 26
%e A360589 12 3750 1.1.4 99 33
%e A360589 ...
%t A360589 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]]
%Y A360589 Cf. A007947, A055932, A126706, A355432, A360768, A360912, A364702.
%K A360589 nonn
%O A360589 1,2
%A A360589 _Michael De Vlieger_, Feb 22 2023