A360765 Numbers k that are neither prime powers nor squarefree, such that A007947(k) * A053669(k) < k.

36, 40, 45, 48, 50, 54, 56, 63, 72, 75, 80, 88, 96, 98, 99, 100, 104, 108, 112, 117, 135, 136, 144, 147, 152, 153, 160, 162, 171, 175, 176, 184, 189, 192, 196, 200, 207, 208, 216, 224, 225, 232, 240, 242, 245, 248, 250, 252, 261, 270, 272, 275, 279, 280, 288, 294, 296, 297, 300, 304, 315, 320, 324, 325

Offset

1,1

Comments

Let rad(k) = A007947(k), and let q = A053669(k).

Let j = A007947(k)*A053669(k) = rad(k)*q.

Composite prime powers p^e such that e > 1 and p^e > 4 have the property j < k. With rad(p^e) = p, in the case of p = 2, pq = 6, 6 < 2^e for e > 2. In the case of odd p, we have 2p < p^e for e > 1.

Squarefree k do not have this property, since rad(k) = k, thus, kq > k by definition of prime q.

For k in this sequence, omega(j) > omega(k), but Omega(j) <= Omega(k), where omega(n) = A001221(n), and Omega(n) = A001222(n).

Subset of A126706.

Links

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Michael De Vlieger, 1016 X 1016 pixel bitmap of n, n = 1..1032256, showing n in black if A126706(n) is in this sequence, else white.

Example

k = 12 is not in the sequence since rad(k)*q(k) = 6*5 = 30, and 30 exceeds k. 18 and 24 are also not in the sequence for the same reason.
k = 36 is in the sequence since rad(36)*q(36) = 6*5 = 30, and 30 < 36.
k = 45 is in the sequence since rad(45)*q(45) = 15*2 = 30, and 30 < 45.

Mathematica

rad[n_] := rad[n] = Times @@ FactorInteger[n][[All, 1]];
q[n_] := If[OddQ[n], 2, p = 2; While[Divisible[n, p], p = NextPrime[p]]; p];
Select[Select[Range[325], Nor[PrimePowerQ[#], SquareFreeQ[#]] &], rad[#]*q[#] < # &] (* Michael De Vlieger, Mar 05 2023 *)

Crossrefs

Cf. A001221, A001222, A007947, A053669, A126706, A246547.

Sequence in context: A349705 A129288 A083248 * A077090 A254836 A067672

Adjacent sequences: A360762 A360763 A360764 * A360766 A360767 A360768

Keyword

nonn

Author

Michael De Vlieger, Mar 05 2023

Status

approved