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A362045 a(n) = smallest k such that k > m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).

%I A362045 #8 Apr 19 2023 19:00:38
%S A362045 48,125,224,243,567,512,832,960,1331,2048,1715,2048,2187,1792,2944,
%T A362045 4131,3125,4617,3712,3968,8125,4374,5589,5000,8192,9317,6144,8192,
%U A362045 10625,8192,19683,15379,19683,12032,11875,11016,11907,13568,12500,19683,13122,14375,15104,16807,15616,19683,19683,17576,45619
%N A362045 a(n) = smallest k such that k > m^2 and rad(k) | m, where rad(k) = A007947(k) and m = A120944(n).
%C A362045 The smallest k such that k > p^2 such that p is prime and rad(k) | p is p^3.
%H A362045 Michael De Vlieger, <a href="/A362045/b362045.txt">Table of n, a(n) for n = 1..32768</a>
%H A362045 Michael De Vlieger, <a href="/A362045/a362045.png">Scatterplot of a(n), m^2, and b(n)</a>, n = 1..2^14, where b(n) = A362044(n) is shown in blue, m^2 in black, and a(n) in red.
%e A362045 a(1) = 48 since m = 6 and the smallest k > m^2 such that rad(k) | 6 is 48. This is to say, the number that follows 6^2 in A003586 is 48.
%e A362045 a(2) = 80 since m = 10 and the smallest k > m^2 such that rad(k) | 10 is 125. This is to say, the number that precedes 10^2 in A003592 is 125.
%e A362045 Table of n = 1..12, m = A120944(n), m^2, and a(n).
%e A362045    n    m    m^2   a(n)
%e A362045   ---------------------
%e A362045    1    6     36     48
%e A362045    2   10    100    125
%e A362045    3   14    196    224
%e A362045    4   15    225    243
%e A362045    5   21    441    567
%e A362045    6   22    484    512
%e A362045    7   26    676    832
%e A362045    8   30    900    960
%e A362045    9   33   1089   1331
%e A362045   10   34   1156   2048
%e A362045   11   35   1225   1715
%e A362045   12   38   1444   2048
%t A362045 Table[m = k^2 + 1; While[! Divisible[k, Times @@ FactorInteger[m][[All, 1]]], m++]; m, {k, Select[Range[6, 133], And[CompositeQ[#], SquareFreeQ[#]] &]}]
%Y A362045 Cf. A007947, A120944, A289280, A362044.
%K A362045 nonn
%O A362045 1,1
%A A362045 _Michael De Vlieger_, Apr 05 2023