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A362432 a(n) is the smallest k > A126706(n) such that rad(k) = rad(A126706(n)) and k mod n != 0, where rad(n) = A007947(n).

%I A362432 #15 May 20 2023 15:04:56
%S A362432 18,24,50,36,98,48,50,242,75,54,80,338,72,98,90,147,578,96,135,722,
%T A362432 100,126,242,120,1058,108,112,363,160,338,144,196,1682,507,150,1922,
%U A362432 168,198,225,578,350,162,189,2738,180,722,867,234,200,192,3362,252,1083,3698,245,242,240,1058,4418,441,216,224
%N A362432 a(n) is the smallest k > A126706(n) such that rad(k) = rad(A126706(n)) and k mod n != 0, where rad(n) = A007947(n).
%C A362432 Let m = A126706(n) and r = rad(m).
%C A362432 Smallest number k greater than m that shares the same squarefree kernel as m, yet does not divide m.
%C A362432 a(n) is in A126706, not a permutation of A126706.
%C A362432 k/r and m/r are coprime.
%C A362432 a(n) < m^2, since k/m < r.
%H A362432 Michael De Vlieger, <a href="/A362432/b362432.txt">Table of n, a(n) for n = 1..10000</a>
%H A362432 Michael De Vlieger, <a href="/A362432/a362432.png">Log log scatterplot of a(n)</a>, n = 1..2^12, showing records in red.
%e A362432 A126706(1) = 12; the smallest k > 12 such that both rad(k) = rad(12) = 6 and 12 does not divide k is a(1) = 18.
%e A362432 A126706(2) = 18; the smallest k > 18 such that both rad(k) = rad(18) = 6 and 18 does not divide k is a(2) = 24.
%e A362432 A126706(3) = 20; the smallest k > 20 such that rad(k) = rad(20) = 10, indivisible by 20, is a(3) = 50.
%e A362432 A126706(7) = 40; the smallest k > 40 such that rad(k) = rad(40) = 10, indivisible by 40, is a(7) = 50.
%t A362432 rad[x_] := rad[x] = Times @@ FactorInteger[x][[All, 1]]; Table[k = m + 1; Function[r, While[Nand[rad[k] == r, ! Divisible[k, m]], k++]][rad[m]]; k, {m, Select[Range[196], Nor[PrimePowerQ[#], SquareFreeQ[#]] &]}]
%Y A362432 Cf. A007947, A065642, A126706.
%K A362432 nonn
%O A362432 1,1
%A A362432 _Michael De Vlieger_, May 19 2023