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A362044 a(n) = largest k such that k < m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

%I A362044 #11 Apr 19 2023 19:00:42
%S A362044 32,80,128,135,343,352,512,864,891,1088,875,1216,1053,1728,2048,2187,
%T A362044 1375,2187,2048,2048,3125,4224,2187,4802,4736,3773,5832,5248,4913,
%U A362044 5504,7047,4459,7533,8192,6859,10368,10935,8192,11264,8991,12312,12167,8192,5831,8192,9963,10449,16640,16807,17152,18432
%N A362044 a(n) = largest k such that k < m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).
%C A362044 The largest k such that k < p^2 such that p is prime and rad(k) | p is p itself.
%H A362044 Michael De Vlieger, <a href="/A362044/b362044.txt">Table of n, a(n) for n = 1..10000</a>
%H A362044 Michael De Vlieger, <a href="/A362044/a362044.png">Scatterplot of a(n), m^2, and b(n)</a>, n = 1..2^14, where b(n) = A362045(n) is shown in red, m^2 in black, and a(n) in blue.
%e A362044 a(1) = 32 since m = 6 and the largest k < m^2 such that rad(k) | 6 is 32. This is to say, the number that precedes 6^2 in A003586 is 32.
%e A362044 a(2) = 80 since m = 10 and the largest k < m^2 such that rad(k) | 10 is 80. This is to say, the number that precedes 10^2 in A003592 is 80.
%e A362044 Table of n = 1..12, m = A120944(n), a(n), and m^2.
%e A362044    n    m    a(n)   m^2
%e A362044   ---------------------
%e A362044    1    6     32     36
%e A362044    2   10     80    100
%e A362044    3   14    128    196
%e A362044    4   15    135    225
%e A362044    5   21    343    441
%e A362044    6   22    352    484
%e A362044    7   26    512    676
%e A362044    8   30    864    900
%e A362044    9   33    891   1089
%e A362044   10   34   1088   1156
%e A362044   11   35    875   1225
%e A362044   12   38   1216   1444
%t A362044 Table[m = k^2 - 1; While[! Divisible[k, Times @@ FactorInteger[m][[All, 1]]], m--]; m, {k, Select[Range[6, 133], And[CompositeQ[#], SquareFreeQ[#]] &]}]
%Y A362044 Cf. A007947, A120944, A289280, A362045.
%K A362044 nonn
%O A362044 1,1
%A A362044 _Michael De Vlieger_, Apr 05 2023