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

24, 50, 54, 60, 75, 100, 102, 108, 112, 126, 165, 168, 170, 174, 180, 186, 189, 190, 192, 198, 200, 204, 216, 225, 231, 238, 242, 245, 315, 340, 363, 370, 374, 390, 396, 400, 402, 405, 408, 414, 416, 420, 426, 429, 432, 435, 442, 462, 465, 476, 480, 484, 490, 492

Offset: 1

Author

Michael De Vlieger, Sep 22 2024

Keywords

• nonn

Comments

The term powerful used here refers to k in A001694, and rad = A007947.

Includes m such that the largest k = A373736(m) in row m of A272618 is not in A001694.

Subset of A024619, since for prime powers m = p^e, e >= 1, all k <= m such that rad(k) | m also divide m.

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.

Examples

			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.
24 is included since nondivisor 18 = 2 * 3^2 is such that rad(18) | 24 and is not powerful.
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.
60 is in the sequence because nondivisor 54 = 2 * 3^3 but rad(54) | 60 and 54 is not powerful, etc.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..16464

Crossrefs

Cf. A001694, A007947, A024619, A246547, A272618, A286708, A373736, A376421.

Programs

Mathematica

rad[x_] := Times @@ FactorInteger[x][[All, 1]];
Table[If[PrimePowerQ[n], Nothing,
  If[! Divisible[#, rad[#]^2], n, Nothing] &@
   SelectFirst[Range[n - 1, 1, -1],
    And[! Divisible[n, #], Divisible[n, rad[#]]] &] ], {n, 2, 500}]