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A376422 Numbers m with largest nondivisor k <= m such that rad(k) | m is not powerful, where rad = A007497.

%I A376422 #6 Sep 23 2024 13:37:48
%S A376422 24,50,54,60,75,100,102,108,112,126,165,168,170,174,180,186,189,190,
%T A376422 192,198,200,204,216,225,231,238,242,245,315,340,363,370,374,390,396,
%U A376422 400,402,405,408,414,416,420,426,429,432,435,442,462,465,476,480,484,490,492
%N A376422 Numbers m with largest nondivisor k <= m such that rad(k) | m is not powerful, where rad = A007497.
%C A376422 The term powerful used here refers to k in A001694, and rad = A007947.
%C A376422 Includes m such that the largest k = A373736(m) in row m of A272618 is not in A001694.
%C A376422 Subset of A024619, since for prime powers m = p^e, e >= 1, all k <= m such that rad(k) | m also divide m.
%C A376422 Subset of A376421, since nondivisor k such that rad(k) | m must be composite, and composite prime powers m in A246547 are a subset of A001694.
%H A376422 Michael De Vlieger, <a href="/A376422/b376422.txt">Table of n, a(n) for n = 1..16464</a>
%e A376422 6 is not included since nondivisor 4 = 2^2 is such that rad(4) | 6, but 4 is powerful since it is a perfect power of a prime.
%e A376422 24 is included since nondivisor 18 = 2 * 3^2 is such that rad(18) | 24 and is not powerful.
%e A376422 42 is not included since nondivisor 36 = 2^2 * 3^2 is such that rad(36) | 42 but 36 is powerful, since all exponents of prime power factors of 36 exceed 1, i.e., 36 is in A286708, a subset of A001694.
%e A376422 60 is in the sequence because nondivisor 54 = 2 * 3^3 but rad(54) | 60 and 54 is not powerful, etc.
%t A376422 rad[x_] := Times @@ FactorInteger[x][[All, 1]];
%t A376422 Table[If[PrimePowerQ[n], Nothing,
%t A376422   If[! Divisible[#, rad[#]^2], n, Nothing] &@
%t A376422    SelectFirst[Range[n - 1, 1, -1],
%t A376422     And[! Divisible[n, #], Divisible[n, rad[#]]] &] ], {n, 2, 500}]
%Y A376422 Cf. A001694, A007947, A024619, A246547, A272618, A286708, A373736, A376421.
%K A376422 nonn
%O A376422 1,1
%A A376422 _Michael De Vlieger_, Sep 22 2024